This document discusses curvilinear motion and kinematics. It introduces position vectors, path coordinates, velocity vectors, and acceleration vectors for particles moving in three-dimensional space. Key concepts covered include defining the position vector r(t) from a reference point to the particle, the instantaneous velocity vector v as the time derivative of r(t), and the acceleration vector a as the time derivative of v. When working in Cartesian coordinates, the derivatives of vector components are simply the derivatives of the individual x, y, z components.
The document discusses degrees of freedom, kinematic constraints, and rectilinear motion. It defines degrees of freedom as the number of coordinates needed to specify the position of a body. Kinematic constraints reduce degrees of freedom by limiting dimensions or relationships between coordinates. Rectilinear motion describes the position, velocity, and acceleration of a particle moving in a straight line over time.
The document provides an overview of key physics equations and concepts for Form 4 students, including equations for relative deviation, prefixes, units for area and volume, equations for average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key terms are defined for important concepts like displacement, time, mass, force, and velocity. Formulas are presented for calculations involving these fundamental physics quantities and relationships.
This document discusses linear motion, forces, and momentum. It includes:
1) Equations for linear motion including displacement, velocity, acceleration, and kinematic equations.
2) Descriptions of linear motion graphs including how to determine velocity and acceleration from displacement-time, velocity-time, and acceleration-time graphs.
3) Definitions of inertia, Newton's First Law of Motion, and momentum as the product of mass and velocity.
4) The principle of conservation of momentum and descriptions of elastic, inelastic, and explosive collisions.
This document summarizes key principles of Einstein's theory of special relativity, including:
1) Special relativity is based on two postulates - the principle of relativity and the constant speed of light. This leads to time dilation, length contraction, and the equivalence of mass and energy.
2) Lorentz transformations relate the coordinates of observers in different inertial reference frames and explain how the speed of light remains constant for all observers.
3) Spacetime diagrams illustrate how events that are simultaneous for one observer may not be for another due to the finite speed of light and relativity of simultaneity.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
1. The document defines various physics concepts including units of measurement, kinematics equations, forces and motion, energy, heat, waves, and optics.
2. Prefixes are provided for the standard form of various units including tera, giga, mega, kilo, deci, centi, and milli.
3. Formulas are given for average speed, velocity, acceleration, linear motion, momentum, impulse, work, power, and more.
4. Concepts around pressure, density, buoyancy, heat transfer, the gas laws, refraction, lenses, and telescopes are also summarized.
Kinematics of a Particle document discusses:
1) Kinematics involves describing motion without considering forces, studying how position, velocity, and acceleration change over time for a particle.
2) Rectilinear motion involves a particle moving along a straight line, where position (x) is defined as the distance from a fixed origin, velocity (v) is the rate of change of position over time, and acceleration (a) is the rate of change of velocity over time.
3) Examples are provided to demonstrate solving kinematics problems using differentiation, integration, and relationships between position, velocity, acceleration graphs. Problems involve determining velocity, acceleration, distance or displacement given various relationships between these quantities.
This document discusses rectilinear motion and concepts related to position, velocity, speed, and acceleration for objects moving along a straight line. It defines velocity as the rate of change of position with respect to time and speed as the magnitude of velocity. Acceleration is defined as the rate of change of velocity with respect to time. Examples of position, velocity, speed, and acceleration graphs are shown for an object with the position function s(t) = t^3 - 6t^2. The document also analyzes position versus time graphs to determine characteristics of an object's motion at different points. Finally, it works through an example of analyzing the motion of a particle with position function s(t) = 2t^3 -
The document discusses degrees of freedom, kinematic constraints, and rectilinear motion. It defines degrees of freedom as the number of coordinates needed to specify the position of a body. Kinematic constraints reduce degrees of freedom by limiting dimensions or relationships between coordinates. Rectilinear motion describes the position, velocity, and acceleration of a particle moving in a straight line over time.
The document provides an overview of key physics equations and concepts for Form 4 students, including equations for relative deviation, prefixes, units for area and volume, equations for average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key terms are defined for important concepts like displacement, time, mass, force, and velocity. Formulas are presented for calculations involving these fundamental physics quantities and relationships.
This document discusses linear motion, forces, and momentum. It includes:
1) Equations for linear motion including displacement, velocity, acceleration, and kinematic equations.
2) Descriptions of linear motion graphs including how to determine velocity and acceleration from displacement-time, velocity-time, and acceleration-time graphs.
3) Definitions of inertia, Newton's First Law of Motion, and momentum as the product of mass and velocity.
4) The principle of conservation of momentum and descriptions of elastic, inelastic, and explosive collisions.
This document summarizes key principles of Einstein's theory of special relativity, including:
1) Special relativity is based on two postulates - the principle of relativity and the constant speed of light. This leads to time dilation, length contraction, and the equivalence of mass and energy.
2) Lorentz transformations relate the coordinates of observers in different inertial reference frames and explain how the speed of light remains constant for all observers.
3) Spacetime diagrams illustrate how events that are simultaneous for one observer may not be for another due to the finite speed of light and relativity of simultaneity.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
1. The document defines various physics concepts including units of measurement, kinematics equations, forces and motion, energy, heat, waves, and optics.
2. Prefixes are provided for the standard form of various units including tera, giga, mega, kilo, deci, centi, and milli.
3. Formulas are given for average speed, velocity, acceleration, linear motion, momentum, impulse, work, power, and more.
4. Concepts around pressure, density, buoyancy, heat transfer, the gas laws, refraction, lenses, and telescopes are also summarized.
Kinematics of a Particle document discusses:
1) Kinematics involves describing motion without considering forces, studying how position, velocity, and acceleration change over time for a particle.
2) Rectilinear motion involves a particle moving along a straight line, where position (x) is defined as the distance from a fixed origin, velocity (v) is the rate of change of position over time, and acceleration (a) is the rate of change of velocity over time.
3) Examples are provided to demonstrate solving kinematics problems using differentiation, integration, and relationships between position, velocity, acceleration graphs. Problems involve determining velocity, acceleration, distance or displacement given various relationships between these quantities.
This document discusses rectilinear motion and concepts related to position, velocity, speed, and acceleration for objects moving along a straight line. It defines velocity as the rate of change of position with respect to time and speed as the magnitude of velocity. Acceleration is defined as the rate of change of velocity with respect to time. Examples of position, velocity, speed, and acceleration graphs are shown for an object with the position function s(t) = t^3 - 6t^2. The document also analyzes position versus time graphs to determine characteristics of an object's motion at different points. Finally, it works through an example of analyzing the motion of a particle with position function s(t) = 2t^3 -
The document provides an overview of key physics equations and concepts related to forces and motion, including equations for relative deviation, prefixes, units of area and volume, average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key variables and their units are defined for each equation. Examples of displacement-time and velocity-time graphs are also included to illustrate the relationships between displacement, velocity, time, and acceleration.
This document discusses kinematics, which is the geometry of motion without considering forces. It defines key concepts like displacement, velocity, acceleration, and their relationships. It presents four kinematic equations and provides examples of using these equations and graphs of position-time and velocity-time to solve kinematics problems for objects undergoing uniform and non-uniform acceleration.
1) The document discusses motion in one dimension, including speed, velocity, acceleration, and formulas for constant acceleration.
2) It defines key terms like speed, velocity, average velocity, instantaneous velocity, acceleration, and average versus instantaneous acceleration.
3) Examples are provided of situations with constant acceleration, including gravitational acceleration near Earth's surface of 9.8 m/s2.
This document provides an overview of key physics concepts related to kinematics including:
- Vectors and scalars
- Displacement, distance, velocity, acceleration, and their relationships
- Mass vs weight
- Motion graphs including position, velocity, and acceleration graphs
- Kinematics equations for constant acceleration including relationships between displacement, velocity, acceleration, and time
- Sample kinematics problems and explanations of concepts like uniform acceleration are provided.
1. The document discusses various kinematics problems involving motion under uniform acceleration. It provides solutions using graphical, analytical and vector methods.
2. Methods include calculating time taken to cross a river based on velocities and angles, determining average and instantaneous velocities from distance-time graphs, resolving velocities into components, and finding the distance between particles moving with different initial velocities.
3. One problem involves three particles moving in a circle such that they are always at the vertices of an equilateral triangle, and calculates the distance traveled by one particle before they meet.
This document provides an overview of kinematic equations and how to approach solving kinematics problems. It includes:
- A list of common kinematic equations and how they are used depending on whether an object starts from rest, has constant velocity, or constant acceleration.
- Guidance on identifying key variables like displacement, velocity, acceleration and determining the correct signs based on the problem context.
- Worked examples showing how to set up and solve kinematics problems step-by-step using the appropriate equations.
- Tips for dealing with problems that may require using multiple equations or are multi-stage problems where acceleration changes. The key is to solve for available variables and use those solutions to determine missing variables
Chapter 12 kinematics of a particle part-iSajid Yasin
This document provides information about a dynamics course, including:
- The instructor is Engr. Sajid Yasin from the Department of Mechanical Technology at MNS UET Multan.
- Lecture times are Tuesday from 7:30-9:30 PM in Room F5 and Wednesday lab from 6:30-9:30 PM in D2.
- Required textbooks and the method of assessment including exams, quizzes, assignments, and attendance are listed.
The document discusses particle kinematics and concepts such as displacement, velocity, acceleration, and their relationships for rectilinear and curvilinear motion. Key concepts covered include definitions of displacement, average and instantaneous velocity, acceleration, graphical representations of position, velocity, and acceleration over time, and analytical methods for solving kinematic equations involving constant or variable acceleration. Several sample problems are provided to illustrate applying these kinematic concepts and relationships to solve for variables like time, velocity, acceleration, and displacement given relevant conditions.
This document provides solutions to 24 problems in special relativity from an undergraduate physics textbook. It was created by Charles Asman, Adam Monahan and Malcolm McMillan at the University of British Columbia for their physics students. The problems cover various topics in special relativity including time dilation, length contraction, relativistic Doppler shift, and Lorentz transformations. Standard frames of reference and equations are defined. Detailed step-by-step solutions are provided for each problem.
1) The document discusses key concepts in physics including position, displacement, velocity, acceleration, and how to interpret graphs related to these concepts. It provides definitions, formulas, example problems and explanations.
2) Position refers to an object's location relative to a reference point, while displacement describes the change in position. Velocity is the rate of change of position (or displacement divided by time) and acceleration is the rate of change of velocity.
3) Graphs are presented that relate position, velocity, displacement and acceleration to time, with the slope of certain graphs representing velocity or acceleration. Example problems are worked through to illustrate these concepts.
it is an advance level presentation, of o level.includes topics such as velocity,acceleration,and detail briefings , hope so students may gain benefit.
This document contains Homer Reid's solutions to problems from Goldstein's Classical Mechanics textbook. The first problem solved involves a radioactive nucleus decaying and emitting an electron and neutrino. The solution finds the direction and momentum of the recoiling nucleus. The next problems solved include deriving the escape velocity of Earth, the equation of motion for a rocket, relating kinetic energy to momentum for varying mass systems, and several other kinematic and constraint problems.
Whole Procedure of Equations of motion.Nafria_duky
The document discusses equations of motion and provides supporting details. It begins by listing 5 equations that satisfy the conditions to be considered equations of motion. These are derived from the velocity-time graph and include displacement-independent, velocity-independent, time-independent, initial velocity-independent, and acceleration-independent equations. The document also notes that all problems can be solved using just 2 of these 5 equations and defends the claim that a fourth equation, S=vt-1/2at^2, was first derived by the author in 2001. It provides background on the author's efforts over many years to have this recognized.
The document discusses equations for calculating velocity, acceleration, displacement, and time for objects moving with constant velocity or uniform acceleration. It provides the key equations:
1) For constant velocity, average velocity (v) equals displacement (Δx) over time (Δt), and displacement equals velocity times time.
2) For uniform acceleration, final velocity (vf) equals initial velocity (v0) plus acceleration (a) times time (Δt), and displacement equals initial velocity times time plus one-half acceleration times time squared.
3) A single equation relates displacement, initial velocity, final velocity, and acceleration, which can be rearranged to solve for any of those variables.
Serway, raymond a physics for scientists and engineers (6e) solutionsTatiani Andressa
This document contains a chapter outline and sample questions and solutions for a physics and measurement chapter. The chapter outline lists topics like standards of length, mass and time, density and atomic mass, and dimensional analysis. The questions and solutions provide examples of calculations involving converting between units, determining densities, and applying dimensional analysis.
Ekeeda Provides Online Civil Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
The document discusses kinematics in one dimension, including definitions of distance, displacement, speed, velocity, acceleration, and how to determine the signs of displacement, velocity, and acceleration. It provides examples of how to solve problems involving initial and final velocity, acceleration, displacement, and time for objects undergoing uniform acceleration. Key concepts covered include average and instantaneous velocity, graphical analysis of position vs time and velocity vs time graphs, and formulas for constant acceleration including the relationships between displacement, initial/final velocities, time, and acceleration.
1. This document contains solutions to problems from Chapter 1 of the textbook "The Physics of Vibrations and Waves".
2. It provides detailed calculations and explanations for problems related to simple harmonic motion, including determining restoring forces, stiffness, frequencies, and solving differential equations of motion.
3. Examples include a simple pendulum, a mass on a spring, and vibrations of strings, membranes, and gas columns.
This document introduces key concepts of 1-D kinematics including:
1) Distance refers to the total path traveled while displacement refers to the change in position from initial to final point.
2) Kinematics describes motion without considering causes, and 1-D kinematics refers to motion in a straight line.
3) Graphs can distinguish between distance and displacement, with displacement being the straight line segment between initial and final points.
4) Key concepts such as average speed, average velocity, acceleration, and free fall are introduced along with relevant equations and demonstrations.
Newton™s Laws; Moment of a Vector; Gravitation; Finite Rotations; Trajectory of a Projectile with Air Resistance; The Simple Pendulum; The Linear Harmonic Oscillator; The Damped Harmonic Oscillator
The document provides suggestions for self-care activities during menstruation including relaxation, yoga, aromatherapy, natural face treatments, and creating a spa-like atmosphere at home. It also outlines content marketing strategies like creating blog posts, videos, and social media updates around these self-care topics to engage customers and tie partnerships with related non-profits or local businesses.
The document provides an overview of key physics equations and concepts related to forces and motion, including equations for relative deviation, prefixes, units of area and volume, average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key variables and their units are defined for each equation. Examples of displacement-time and velocity-time graphs are also included to illustrate the relationships between displacement, velocity, time, and acceleration.
This document discusses kinematics, which is the geometry of motion without considering forces. It defines key concepts like displacement, velocity, acceleration, and their relationships. It presents four kinematic equations and provides examples of using these equations and graphs of position-time and velocity-time to solve kinematics problems for objects undergoing uniform and non-uniform acceleration.
1) The document discusses motion in one dimension, including speed, velocity, acceleration, and formulas for constant acceleration.
2) It defines key terms like speed, velocity, average velocity, instantaneous velocity, acceleration, and average versus instantaneous acceleration.
3) Examples are provided of situations with constant acceleration, including gravitational acceleration near Earth's surface of 9.8 m/s2.
This document provides an overview of key physics concepts related to kinematics including:
- Vectors and scalars
- Displacement, distance, velocity, acceleration, and their relationships
- Mass vs weight
- Motion graphs including position, velocity, and acceleration graphs
- Kinematics equations for constant acceleration including relationships between displacement, velocity, acceleration, and time
- Sample kinematics problems and explanations of concepts like uniform acceleration are provided.
1. The document discusses various kinematics problems involving motion under uniform acceleration. It provides solutions using graphical, analytical and vector methods.
2. Methods include calculating time taken to cross a river based on velocities and angles, determining average and instantaneous velocities from distance-time graphs, resolving velocities into components, and finding the distance between particles moving with different initial velocities.
3. One problem involves three particles moving in a circle such that they are always at the vertices of an equilateral triangle, and calculates the distance traveled by one particle before they meet.
This document provides an overview of kinematic equations and how to approach solving kinematics problems. It includes:
- A list of common kinematic equations and how they are used depending on whether an object starts from rest, has constant velocity, or constant acceleration.
- Guidance on identifying key variables like displacement, velocity, acceleration and determining the correct signs based on the problem context.
- Worked examples showing how to set up and solve kinematics problems step-by-step using the appropriate equations.
- Tips for dealing with problems that may require using multiple equations or are multi-stage problems where acceleration changes. The key is to solve for available variables and use those solutions to determine missing variables
Chapter 12 kinematics of a particle part-iSajid Yasin
This document provides information about a dynamics course, including:
- The instructor is Engr. Sajid Yasin from the Department of Mechanical Technology at MNS UET Multan.
- Lecture times are Tuesday from 7:30-9:30 PM in Room F5 and Wednesday lab from 6:30-9:30 PM in D2.
- Required textbooks and the method of assessment including exams, quizzes, assignments, and attendance are listed.
The document discusses particle kinematics and concepts such as displacement, velocity, acceleration, and their relationships for rectilinear and curvilinear motion. Key concepts covered include definitions of displacement, average and instantaneous velocity, acceleration, graphical representations of position, velocity, and acceleration over time, and analytical methods for solving kinematic equations involving constant or variable acceleration. Several sample problems are provided to illustrate applying these kinematic concepts and relationships to solve for variables like time, velocity, acceleration, and displacement given relevant conditions.
This document provides solutions to 24 problems in special relativity from an undergraduate physics textbook. It was created by Charles Asman, Adam Monahan and Malcolm McMillan at the University of British Columbia for their physics students. The problems cover various topics in special relativity including time dilation, length contraction, relativistic Doppler shift, and Lorentz transformations. Standard frames of reference and equations are defined. Detailed step-by-step solutions are provided for each problem.
1) The document discusses key concepts in physics including position, displacement, velocity, acceleration, and how to interpret graphs related to these concepts. It provides definitions, formulas, example problems and explanations.
2) Position refers to an object's location relative to a reference point, while displacement describes the change in position. Velocity is the rate of change of position (or displacement divided by time) and acceleration is the rate of change of velocity.
3) Graphs are presented that relate position, velocity, displacement and acceleration to time, with the slope of certain graphs representing velocity or acceleration. Example problems are worked through to illustrate these concepts.
it is an advance level presentation, of o level.includes topics such as velocity,acceleration,and detail briefings , hope so students may gain benefit.
This document contains Homer Reid's solutions to problems from Goldstein's Classical Mechanics textbook. The first problem solved involves a radioactive nucleus decaying and emitting an electron and neutrino. The solution finds the direction and momentum of the recoiling nucleus. The next problems solved include deriving the escape velocity of Earth, the equation of motion for a rocket, relating kinetic energy to momentum for varying mass systems, and several other kinematic and constraint problems.
Whole Procedure of Equations of motion.Nafria_duky
The document discusses equations of motion and provides supporting details. It begins by listing 5 equations that satisfy the conditions to be considered equations of motion. These are derived from the velocity-time graph and include displacement-independent, velocity-independent, time-independent, initial velocity-independent, and acceleration-independent equations. The document also notes that all problems can be solved using just 2 of these 5 equations and defends the claim that a fourth equation, S=vt-1/2at^2, was first derived by the author in 2001. It provides background on the author's efforts over many years to have this recognized.
The document discusses equations for calculating velocity, acceleration, displacement, and time for objects moving with constant velocity or uniform acceleration. It provides the key equations:
1) For constant velocity, average velocity (v) equals displacement (Δx) over time (Δt), and displacement equals velocity times time.
2) For uniform acceleration, final velocity (vf) equals initial velocity (v0) plus acceleration (a) times time (Δt), and displacement equals initial velocity times time plus one-half acceleration times time squared.
3) A single equation relates displacement, initial velocity, final velocity, and acceleration, which can be rearranged to solve for any of those variables.
Serway, raymond a physics for scientists and engineers (6e) solutionsTatiani Andressa
This document contains a chapter outline and sample questions and solutions for a physics and measurement chapter. The chapter outline lists topics like standards of length, mass and time, density and atomic mass, and dimensional analysis. The questions and solutions provide examples of calculations involving converting between units, determining densities, and applying dimensional analysis.
Ekeeda Provides Online Civil Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
The document discusses kinematics in one dimension, including definitions of distance, displacement, speed, velocity, acceleration, and how to determine the signs of displacement, velocity, and acceleration. It provides examples of how to solve problems involving initial and final velocity, acceleration, displacement, and time for objects undergoing uniform acceleration. Key concepts covered include average and instantaneous velocity, graphical analysis of position vs time and velocity vs time graphs, and formulas for constant acceleration including the relationships between displacement, initial/final velocities, time, and acceleration.
1. This document contains solutions to problems from Chapter 1 of the textbook "The Physics of Vibrations and Waves".
2. It provides detailed calculations and explanations for problems related to simple harmonic motion, including determining restoring forces, stiffness, frequencies, and solving differential equations of motion.
3. Examples include a simple pendulum, a mass on a spring, and vibrations of strings, membranes, and gas columns.
This document introduces key concepts of 1-D kinematics including:
1) Distance refers to the total path traveled while displacement refers to the change in position from initial to final point.
2) Kinematics describes motion without considering causes, and 1-D kinematics refers to motion in a straight line.
3) Graphs can distinguish between distance and displacement, with displacement being the straight line segment between initial and final points.
4) Key concepts such as average speed, average velocity, acceleration, and free fall are introduced along with relevant equations and demonstrations.
Newton™s Laws; Moment of a Vector; Gravitation; Finite Rotations; Trajectory of a Projectile with Air Resistance; The Simple Pendulum; The Linear Harmonic Oscillator; The Damped Harmonic Oscillator
The document provides suggestions for self-care activities during menstruation including relaxation, yoga, aromatherapy, natural face treatments, and creating a spa-like atmosphere at home. It also outlines content marketing strategies like creating blog posts, videos, and social media updates around these self-care topics to engage customers and tie partnerships with related non-profits or local businesses.
El documento contiene varias conversaciones cortas en las que un hombre le pide trabajo a otra persona en repetidas ocasiones, pero se le niega cada vez. En otra conversación, el hombre espera su turno para hablar con alguien, pero no se especifica sobre qué.
Social media allows for communication and sharing between businesses and customers. It comes in many forms like forums, blogs, social networks, multimedia sharing, bookmarking, and microblogging. Companies use social media to market and promote their brand, get customer feedback, and build customer loyalty. Examples of successful businesses that use social media include large companies like Coca-Cola and small businesses alike.
HCL Infosystems launches MyEduWorld, a new educational ecosystem for India encompassing digital curriculum, applications, videos, animations, and quizzes. It is available on Android tablets as MyEduWorld Tab and Windows PCs/laptops as MyEduWorld Drive. The platform allows self-paced learning anywhere and tracks student progress through assessment tests and reports to parents. It aims to make learning more engaging for students.
Nanotechnology refers to engineering at the molecular scale, typically 1-100 nanometers. In its original conception, it meant precisely constructing products through molecular manufacturing. While current work is simpler, the long term vision is to build complex products through mechanochemistry and molecular machine systems. Realizing this advanced nanotechnology could enable a manufacturing revolution within a decade through self-replicating nanofactories, exponentially proliferating production capabilities but also posing risks if not responsibly developed and governed.
Molecular manufacturing (MM) aims to precisely construct complex products atom-by-atom using mechanical chemistry and robotic assembly. Theoretical studies have shown it is possible to build molecular structures like diamond lattices in this way. A small robotic device called a fabricator could use supplied chemicals to manufacture nanoscale shapes and machines. Multiple fabricators could then be combined to create a personal nanofactory (PN) capable of building human-scale products within months. Once achieved, MM would allow highly efficient, precise, and inexpensive production through self-replicating nanofactories making both commercial and military applications very desirable.
A comparison of historical vs current instructional designClemson University
This document compares historical and current instructional design strategies used in the manufacturing industry. It discusses pioneers in the field from the 1900s like Frederick Taylor who developed timed work instructions. Sidney Pressey created early teaching machines, while B.F. Skinner researched programmed instruction. W.E. Deming applied quality control methods and embraced instructional design. The document finds that while methods have impacted design and learning, approaches have changed little since Taylorism. It identifies future trends like inclusion of diverse perspectives and customized instructional materials.
1) A 20-year-old male presented with seizures secondary to a rhabdoid meningioma that was radically surgically removed five years ago without radiation therapy.
2) Over the past six months, he developed progressive left-sided weakness.
3) New imaging showed a lesion in the same location as the previous meningioma, and spectroscopy showed glioma patterns.
4) A stereotactic biopsy found oligodendroglioma on first report but astrocytoma grade II on neuropathology second opinion. Radical surgery with monitoring and awake surgery is recommended, followed by radiotherapy and follow-up.
AngularJS: A framework to make your life easierWilson Mendes
AngularJS is a javascript framework built and maintained by Google engineers Group, it uses HTML as a "template engine", all this in order to provide a complete solution for the client-side of your application. Also has full compatibility with the most used javascript libraries such as jQuery. It's a new concept for developing web apps client-site.
1. The document summarizes mechanics concepts including equilibrium, force-deformation relationships, and compatibility as they relate to statically indeterminate structures.
2. It provides the equations to solve for forces and deformations on a 2D frame with applied load P and determines displacements are a function of material and geometric properties.
3. The summary solves for the displacements uDx and uDy of point D on the frame in terms of the applied load P, bar cross-sectional area A, length L, and angle θ using the equilibrium and compatibility equations.
The document summarizes plane curvilinear motion and projectile motion. It defines key concepts like position vector, velocity, acceleration, and rectangular coordinate analysis. It provides equations to describe velocity and acceleration in x and y directions. Examples are given to demonstrate calculating displacement, velocity, acceleration from given motion equations. The last two examples solve for minimum initial velocity and angles to just clear a fence or pass through a basketball hoop.
1) Intrinsic coordinates include the tangential, normal and binormal components aligned with the particle's velocity vector, the normal vector to the trajectory, and their cross product, respectively. 2) In intrinsic coordinates, the velocity and acceleration vectors take simple forms, with the acceleration having tangential and normal components related to the curvature of the trajectory. 3) Intrinsic coordinates can simplify analysis of problems where motion is constrained to a known trajectory.
Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1
1) The document describes curvilinear motion and how to analyze the motion of objects moving along curved paths using rectangular components.
2) It provides examples of how to determine the velocity and acceleration of planes in formation and a roller coaster car moving along a fixed helical path using their x, y, and z coordinates.
3) The document also gives an example problem solving for the collision point and speeds of two particles moving along curved paths given their position vectors as functions of time.
Vectors have both magnitude and direction, while scalars only have magnitude. Common vector quantities include displacement, velocity, acceleration, force, and electric and magnetic fields. Vector notation allows physical laws to be written compactly. Vectors can be added, subtracted, and multiplied by scalars. The dot product yields a scalar and the cross product yields a vector perpendicular to the two original vectors. Uniform circular motion results in constant speed but centripetal acceleration directed radially inward.
1) The document discusses kinematics of particles, specifically rectilinear and curvilinear motion.
2) Rectilinear motion involves motion along a straight line that can be described using displacement, velocity, acceleration, and differential equations.
3) Curvilinear motion occurs along a curved path in a plane and uses vector analysis to describe position, displacement, and velocity.
This document provides an overview of tensors, including:
- Tensors generalize scalars, vectors, and allow representation of physical laws in a coordinate-independent way.
- Tensor components transform according to changes of basis vectors under coordinate transformations.
- The metric tensor relates vector components and defines raising/lowering of indices.
- The covariant derivative accounts for how tensor components change across coordinate systems due to curvature.
- Tensor calculus, using covariant derivatives, allows formulation of physical laws in a generally covariant way.
This document discusses the kinematics of particles in rectilinear and curvilinear motion. It defines key concepts like position, displacement, velocity, and acceleration for both continuous and erratic rectilinear motion. Examples are provided to demonstrate how to construct velocity-time and acceleration-time graphs from a given position-time graph, and vice versa. The chapter then discusses general curvilinear motion, defining position, displacement, velocity, and acceleration using vector analysis since the curved path is three-dimensional. Fundamental problems and practice problems are also included.
The document discusses the interpretation of wave functions and Schrodinger's equation in quantum mechanics. It proposes that wave functions can be expressed as complex space vectors that rotate on different axes. This allows wave functions like sinusoidal functions to be expressed using Euler's formula and addressed some issues with differentiation. It suggests wave equations can be satisfied when interactions between systems exhibit exponential behavior over time and position, and proposes the wave function solution Ψ=Ae^-mvxi. Further implications and future outlook are discussed.
The document discusses average speed, average velocity, and instantaneous velocity. It defines average speed as the total distance traveled divided by the time interval. Average velocity is a vector quantity that has both magnitude and direction. Instantaneous velocity is the limit of average velocity as the time interval approaches zero. The document uses examples of objects moving in straight lines and elliptical orbits to illustrate these concepts.
Vector integration involves integrating a vector field along a curve or path, with the result being a vector quantity. It is similar to scalar integration but deals with vector-valued functions instead of scalar functions. Line integrals calculate the total of a scalar or vector field along a curve and are used in physics and engineering to model quantities depending on the path taken. There are two main types: scalar line integrals and vector line integrals.
- The binormal vector B(t) is defined as the cross product of the unit tangent vector T(t) and unit normal vector N(t).
- It is proven that B(t) is a unit vector, meaning it has constant length. Its derivative dB/ds is therefore orthogonal to B(t).
- The torsion τ of a space curve is defined as the rate of change of the binormal vector with respect to arc length s, or τ = -dB/ds·N. Torsion measures how much a curve twists as one moves along it.
- For a plane curve, the torsion is always zero since the cross product that defines torsion is equal to the
Central forces are forces that point radially towards or away from a source and depend only on the distance from the source. For a central force, the angular momentum of a particle is conserved, and the particle's motion occurs in a single plane. By defining an effective potential energy, the two-dimensional problem can be reduced to an equivalent one-dimensional problem, simplifying the analysis. Solving the equations of motion yields the particle's position r and angle θ as functions of time t, or r as a function of θ, describing the trajectory.
This document provides information about a dynamics course taught by Professor Nikolai V. Priezjev. The course will cover kinematics and dynamics using the textbook "Vector Mechanics for Engineers: Dynamics" by Beer, Johnston, Mazurek and Cornwell. Kinematics deals with the geometric aspects of motion without forces or moments. The course objectives are to derive relations between position, velocity and acceleration for various motion types using concepts like the s-t graph and rectangular components.
- The document discusses kinematic concepts such as position, displacement, velocity, and acceleration for particles moving along a straight path. It defines these concepts using equations of motion.
- Rectilinear motion is analyzed by creating graphs of position vs. time, velocity vs. time, and acceleration vs. time. The slopes of these graphs are used to define velocity, acceleration, and how they relate to each other.
- Integrals of the kinematic equations are used to determine relationships between position, velocity, acceleration, and time.
1. The document discusses Maxwell's equations and their solutions, which are travelling waves in free space determined by charges and currents that change over time.
2. It then considers a simple case of sources being a very small volume element, much smaller than the wavelength, where the solution takes a spherical wave form.
3. The document derives an expression for the retarded potentials generated by a moving point charge, known as the Lienard-Wiechert potentials. This involves finding the position and time at which the charge emitted the radiation intercepted at a given observation point and time.
This document discusses concepts in mechanics including kinematics, dynamics, and statics. It defines key terms like reference frames, position vectors, displacement, average speed, average velocity, and instantaneous acceleration. It also provides examples of determining trajectory, displacement, velocity, and center of mass for systems of particles.
This document defines and provides examples of parameterized curves in R3. It introduces the concept of a smooth curve as a smooth map from an interval to R3. Regular curves are defined as smooth curves with everywhere nonzero velocity. Examples of regular curves include circles and helices parameterized by t. Reparameterizations are introduced as changing the parameterization of a curve while tracing the same path. The length of a curve is defined using an integral of speed over the parameter. Arc length parameterization reparameterizes a curve so that the parameter is equal to arc length measured from a fixed point.
The document summarizes key concepts from Stephen Covey's book "The 7 Habits of Highly Effective People". It discusses the first three habits: 1) Be Proactive - take responsibility for your life and focus on things within your control. 2) Begin with the End in Mind - develop a personal mission statement and envision your goals. 3) Put First Things First - prioritize important tasks and spend time on high-impact activities to achieve your goals. Effective time management involves focusing on important tasks rather than urgent tasks.
Us navy introduction to helicopter aerodynamics workbook cnatra p-401 [us n...Mohamed Yaser
Here are the answers to the review questions from Chapter 1:
1. Oxygen comprises approximately 21 percent of the earth's atmosphere.
2. Air density changes in direct proportion to pressure and inverse proportion to temperature, altitude, and humidity.
3. d. less dense.
4. pressure altitude
5. temperature and humidity
6. True
7. As temperature increases above standard day conditions, density altitude increases and air density decreases.
8. 4,500 feet
9. 6,600 feet
10. Increases density altitude, which decreases rotor efficiency.
11. Increased density altitude adversely affects both power required and power available. Power required increases due
The document discusses the history and development of artificial intelligence over the past 70 years. It outlines some of the key milestones in AI research from the early work in the 1950s to modern advances in machine learning using neural networks. While progress has been made, fully general human-level artificial intelligence remains an ongoing challenge being worked on by researchers.
Seddon j. basic helicopter aerodynamics [bsp prof. books 1990]Mohamed Yaser
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The 10 natural_laws_of_successful_time_and_life_managementMohamed Yaser
The document summarizes the 10 natural laws of successful time and life management. It discusses how inner peace comes from aligning daily activities with core values. It provides a pyramid model showing the relationship between values, goals, and tasks. The laws discuss controlling events through planning, setting goals beyond one's comfort zone, and managing behavior by examining beliefs and needs. Behavior patterns reflect underlying beliefs, and negative behaviors are overcome by changing incorrect beliefs.
The 10 Natural Laws Of Successful Time And Life ManagementMohamed Yaser
The document outlines 10 natural laws of successful time and life management according to Hyrum W. Smith. The three most important laws are:
1. Your governing values are the foundation of personal fulfillment. Identifying your core values through writing a "personal constitution" allows you to plan your time effectively.
2. When your daily activities reflect your governing values, prioritized in order of importance, you experience inner peace and avoid neglecting what matters most.
3. You control your life by controlling your time. Focus on identifying vital versus urgent tasks and spend maximum time on important priorities rather than what is simply urgent. Managing events according to your values leads to fulfillment.
This document discusses the development of a new type of battery that could revolutionize energy storage. It describes how the battery uses a solid electrolyte material that conducts ions quickly without using liquid electrolytes. This leads to a battery that charges faster, lasts longer and poses less risk of leaks or fires. The solid-state battery is expected to be commercially available within the next five years and could replace lithium-ion batteries in many applications.
The Northrop F-20 Tigershark was a privately developed fighter aircraft intended to compete with the F-16 Fighting Falcon. It began flying in 1982 but failed to secure any orders due to the US relaxing restrictions on F-16 sales. While the F-20 had performance comparable to early F-16 models, its airframe was based on the older F-5 design with limited expansion capabilities. The last prototype is displayed at the California Science Center.
This document lists dates from January 29th to February 28th. It includes columns for important events and tasks but these columns are empty. The document appears to be a monthly calendar intended to log events and tasks but it currently contains no information.
This document provides an introduction and overview of the structural design process for aerospace structures. It discusses that structural integrity is key to prevent failure, and the majority of accidents are due to structural material failure. The course will provide tools to properly design aerospace structures to ensure integrity. It notes that while aircraft structures have been the main focus, the techniques can be generalized to other structures like space structures. The structural design process is then outlined, with the goal being to ensure integrity while minimizing cost, often meaning weight. Key aspects of structural integrity are also defined.
This document provides a summary of basic MATLAB commands organized into sections:
1) Basic scalar, vector, and matrix operations including declaring variables, performing arithmetic, accessing elements, and clearing memory.
2) Using character strings to print output and combine text.
3) Common mathematical operations like exponents, logarithms, trigonometric functions, rounding, and converting between numeric and string formats.
The document demonstrates how to perform essential tasks in MATLAB through examples and explanations of core commands. It serves as an introductory tutorial for newcomers to learn MATLAB's basic functionality.
This document provides an introduction to using MATLAB for numerical methods in chemical engineering. It discusses how computers solve problems by breaking them down into linear algebraic systems that can be represented as matrix equations. While compiled languages like FORTRAN are efficient, MATLAB is better suited for education and small-to-medium projects because it is an interpreted language, allowing interactive use without needing to compile code. MATLAB handles tasks like input/output, variable naming, and graphics internally through pre-compiled routines.
This document provides a tutorial on basic MATLAB commands for creating, manipulating, and operating on vectors and matrices. It describes how to create vectors and matrices, change their entries, perform matrix multiplication and inversion, extract submatrices, and create special matrices like identity and diagonal matrices. Examples are provided to illustrate various commands like eye, inv, backslash, and how to input vectors, matrices, and create M-files for functions and scripts.
This document introduces an alternative theoretical framework to Einstein's theory of special relativity called Realitivistic Relativity 2.0. It proposes that star systems and atomic systems are fundamentally similar, with stars behaving like protons and planets like electrons. The author derived mathematical relationships between celestial objects and their quantum counterparts with high accuracy. This work maps objects in star systems to particles in atoms and vice versa, challenging existing interpretations of physics. It aims to simplify and unite physics through reexamining data from first principles without preconceived theories.
This document summarizes a lecture on forces and moments transmitted by slender members. It defines slender members as long, skinny structural elements like skis, golf clubs, and I-beams. It discusses axial forces that act along the member's axis, shear forces that act in the plane of the member's face, and bending moments. Sign conventions for these internal loads are also defined. An example is provided to demonstrate calculating the internal forces and moments in a beam by setting up free body diagrams at different points.
The document reviews concepts of uniaxial loading, stress, strain, stress-strain relationships, force-displacement relationships, and deformation and displacement in mechanical structures. It provides an example of calculating forces and displacements in a statically determinate truss. The document outlines an algorithm to determine displacements at point B by considering compatibility of displacements for given forces.
1. S. Widnall, J. Peraire
16.07 Dynamics
Fall 2009
Version 2.0
Lecture L4 - Curvilinear Motion. Cartesian Coordinates
We will start by studying the motion of a particle. We think of a particle as a body which has mass,
but has negligible dimensions. Treating bodies as particles is, of course, an idealization which involves an
approximation. This approximation may be perfectly acceptable in some situations and not adequate in
some other cases. For instance, if we want to study the motion of planets, it is common to consider each
planet as a particle. This simplification is not adequate if we wish to study the precession of a gyroscope or
a spinning top.
Kinematics of curvilinear motion
In dynamics we study the motion and the forces that cause, or are generated as a result of, the motion.
Before we can explore these connections we will look first at the description of motion irrespective of the
forces that produce them. This is the domain of kinematics. On the other hand, the connection between
forces and motions is the domain of kinetics and will be the subject of the next lecture.
Position vector and Path
We consider the general situation of a particle moving in a three dimensional space. To locate the position of
a particle in space we need to set up an origin point, O, whose location is known. The position of a particle
A, at time t, can then be described in terms of the position vector, r, joining points O and A. In general,
this particle will not be still, but its position will change in time. Thus, the position vector will be a function
of time, i.e. r(t). The curve in space described by the particle is called the path, or trajectory.
We introduce the path or arc length coordinate, s, which measures the distance traveled by the particle along
the curved path. Note that for the particular case of rectilinear motion (considered in the review notes) the
arc length coordinate and the coordinate, s, are the same.
1
2. Using the path coordinate we can obtain an alternative representation of the motion of the particle. Consider
that we know r as a function of s, i.e. r(s), and that, in addition we know the value of the path coordinate
as a function of time t, i.e. s(t). We can then calculate the speed at which the particle moves on the path
simply as v = s ≡ ds/dt. We also compute the rate of change of speed as at = s = d2 s/dt2 .
˙ ¨
We consider below some motion examples in which the position vector is referred to a fixed cartesian
coordinate system.
Example Motion along a straight line in 2D
Consider for illustration purposes two particles that move along a line defined by a point P and a unit vector
m. We further assume that at t = 0, both particles are at point P . The position vector of the first particle is
given by r 1 (t) = r P + mt = (rP x + mx t)i + (rP y + my t)j, whereas the position vector of the second particle
is given by r 2 (t) = r P + mt2 = (rP x + mx t2 )i + (rP y + my t2 )j.
Clearly the path for these two particles is the same, but the speed at which each particle moves along the
path is different. This is seen clearly if we parameterize the path with the path coordinate, s. That is,
we write r(s) = r P + ms = (rP x + mx s)i + (rP y + my s)j. It is straightforward to verify that s is indeed
the path coordinate i.e. the distance between two points r(s) and r(s + Δs) is equal to Δs. The two
motions introduced earlier simply correspond to two particles moving according to s1 (t) = t and s2 (t) = t2 ,
respectively. Thus, r 1 (t) = r(s1 (t)) and r 2 (t) = r(s2 (t)).
It turns out that, in many situations, we will not have an expression for the path as a function of s. It is
in fact possible to obtain the speed directly from r(t) without the need for an arc length parametrization of
the trajectory.
Velocity Vector
We consider the positions of the particle at two different times t and t + Δt, where Δt is a small increment
of time. Let Δr = r(r + Δt) − r(t), be the displacement vector as shown in the diagram.
2
3. The average velocity of the particle over this small increment of time is
Δr
v ave = ,
Δt
which is a vector whose direction is that of Δr and whose magnitude is the length of Δr divided by Δt. If
Δt is small, then Δr will become tangent to the path, and the modulus of Δr will be equal to the distance
the particle has moved on the curve Δs.
The instantaneous velocity vector is given by
Δr dr(t)
v = lim ≡ ≡r ,
˙ (1)
Δt→0 Δt dt
and is always tangent to the path. The magnitude, or speed, is given by
Δs ds
v = |v | = lim ≡ ≡s.
˙
Δt→0 Δt dt
Acceleration Vector
In an analogous manner, we can define the acceleration vector. Particle A at time t, occupies position
r(t), and has a velocity v(t), and, at time t + Δt, it has position r(t + Δt) = r(t) + Δr, and velocity
v(t + Δt) = v(t) + Δv. Considering an infinitesimal time increment, we define the acceleration vector as the
derivative of the velocity vector with respect to time,
Δv dv d2 r
a = lim ≡ = 2 . (2)
Δt→0 Δt dt dt
We note that the acceleration vector will reflect the changes of velocity in both magnitude and direction.
The acceleration vector will, in general, not be tangent to the trajectory (in fact it is only tangent when the
velocity vector does not change direction). A sometimes useful way to visualize the acceleration vector is to
3
4. translate the velocity vectors, at different times, such that they all have a common origin, say, O� . Then,
the heads of the velocity vector will change in time and describe a curve in space called the hodograph. We
then see that the acceleration vector is, in fact, tangent to the hodograph at every point.
Expressions (1) and (2) introduce the concept of derivative of a vector. Because a vector has both magnitude
and direction, the derivative will be non-zero when either of them changes (see the review notes on
vectors). In general, the derivative of a vector will have a component which is parallel to the vector itself,
and is due to the magnitude change; and a component which is orthogonal to it, and is due to the direction
change.
Note Unit tangent and arc-length parametrization
The unit tangent vector to the curve can be simply calculated as
et = v/v.
It is clear that the tangent vector depends solely on the geometry of the trajectory and not on the speed
at which the particle moves along the trajectory. That is, the geometry of the trajectory determines the
tangent vector, and hence the direction of the velocity vector. How fast the particle moves along the
trajectory determines the magnitude of the velocity vector. This is clearly seen if we consider the arc-length
parametrization of the trajectory r(s). Then, applying the chain rule for differentiation, we have that,
dr dr ds
v= = = et v ,
dt ds dt
where, s = v, and we observe that dr/ds = et . The fact that the modulus of dr/ds is always unity indicates
˙
that the distance traveled, along the path, by r(s), (recall that this distance is measured by the coordinate
s), per unit of s is, in fact, unity!. This is not surprising since by definition the distance between two
neighboring points is ds, i.e. |dr| = ds.
Cartesian Coordinates
When working with fixed cartesian coordinates, vector differentiation takes a particularly simple form. Since
the vectors i, j, and k do not change, the derivative of a vector A(t) = Ax (t)i + Ay (t)j + Az (t)k, is simply
˙ ˙ ˙ ˙
A(t) = Ax (t)i + Ay (t)j + Az (t)k. That is, the components of the derivative vector are simply the derivatives
of the components.
4
5. Thus, if we refer the position, velocity, and acceleration vectors to a fixed cartesian coordinate system, we
have,
r(t) = x(t)i + y(t)j + z(t)k (3)
˙ ˙ ˙
v(t) = vx (t)i + vy (t)j + vz (t)k = x(t)i + y(t)j + z(t)k = r (t)
˙ (4)
a(t) = ˙ ˙ ˙ ˙
ax (t)i + ay (t)j + az (t)k = vx (t)i + vy (t)j + vz (t)k = v (t) (5)
� �
Here, the speed is given by v = vx + vy + vz , and the magnitude of the acceleration is a = a2 + a2 + a2 .
2 2 2
x y z
The advantages of cartesian coordinate systems is that they are simple to use, and that if a is constant, or
a function of time only, we can integrate each component of the acceleration and velocity independently as
shown in the ballistic motion example.
Example Circular Motion
We consider motion of a particle along a circle of radius R at a constant speed v0 . The parametrization of
a circle in terms of the arc length is
s s
r(s) = R cos( )i + R sin( )j .
R R
Since we have a constant speed v0 , we have s = v0 t. Thus,
v0 t v0 t
r(t) = R cos( )i + R sin( )j .
R R
The velocity is
dr(t) v0 t v0 t
v(t) = = −v0 sin( )i + v0 cos( )j ,
dt R R
5
6. which, clearly, has a constant magnitude |v| = v0 . The acceleration is,
dr(t) v2 v0 t v2 v0 t
a(t) = = − 0 cos( )i − 0 sin( )j .
dt R R R R
Note that, the acceleration is perpendicular to the path (in this case it is parallel to r), since the velocity
vector changes direction, but not magnitude.
We can also verify that, from r(s), the unit tangent vector, et , could be computed directly as
dr(s) s s v0 t v0 t
et = = − sin( )i + cos( ) = − sin( )i + cos( )j .
ds R R R R
Example Motion along a helix
The equation r(t) = R cos ti + R sin tj + htk, defines the motion of a particle moving on a helix of radius R,
and pitch 2πh, at a constant speed. The velocity vector is given by
dr
v= = −R sin ti + R cos tj + hk ,
dt
and the acceleration vector is given by,
dv
a= = −R cos ti + −R sin tj .
dt
In order to determine the speed at which the particle moves we simply compute the modulus of the velocity
vector,
� �
v = |v| = R2 sin2 t + R2 cos2 t + h2 = R 2 + h2 .
If we want to obtain the equation of the path in terms of the arc-length coordinate we simply write,
�
ds = |dr| = vdt = R2 + h2 dt .
√
Integrating, we obtain s = s0 + R2 + h2 t, where s0 corresponds to the path coordinate of the particle
at time zero. Substituting t in terms of s, we obtain the expression for the position vector in terms of the
√ √ √
arc-length coordinate. In this case, r(s) = R cos(s/ R2 + h2 )i + R sin(s/ R2 + h2 )j + hs/ R2 + h2 k. The
figure below shows the particle trajectory for R = 1 and h = 0.1.
2
1.5
1
0.5
0
1
0.5 1
0 0.5
0.5 0
0.5
1 1
6
7. Example Ballistic Motion
Consider the free-flight motion of a projectile which is initially launched with a velocity v 0 = v0 cos φi +
v0 sin φj. If we neglect air resistance, the only force on the projectile is the weight, which causes the projectile
to have a constant acceleration a = −gj. In component form this equation can be written as dvx /dt = 0
and dvy /dt = −g. Integrating and imposing initial conditions, we get
vx = v0 cos φ, vy = v0 sin φ − gt ,
where we note that the horizontal velocity is constant. A further integration yields the trajectory
1
x = x0 + (v0 cos φ) t, y = y0 + (v0 sin φ) t − gt2 ,
2
which we recognize as the equation of a parabola.
The maximum height, ymh , occurs when vy (tmh ) = 0, which gives tmh = (v0 /g) sin φ, or,
v0 sin2 φ
2
ymh = y0 + .
2g
The range, xr , can be obtained by setting y = y0 , which gives tr = (2v0 /g) sin φ, or,
2
2v0 sin φ cos φ v 2 sin(2φ)
xr = x0 + = x0 + 0 .
g g
We see that if we want to maximize the range xr , for a given velocity v0 , then sin(2φ) = 1, or φ = 45o .
Finally, we note that if we want to model a more realistic situation and include aerodynamic drag forces of
the form, say, −κv 2 , then we would not be able to solve for x and y independently, and this would make the
problem considerably more complicated (usually requiring numerical integration).
ADDITIONAL READING
J.L. Meriam and L.G. Kraige, Engineering Mechanics, DYNAMICS, 5th Edition
2/1, 2/3, 2/4
7