ch2 - Kinematics in One Dimension Chapter Outline GENERAL...

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Unformatted text preview: Kinematics in One Dimension Chapter Outline GENERAL PHYSICS PHY 302K Reference Frames and Displacement Average and Instantaneous Velocity Chapter 2: Kinematics in One Dimension Acceleration Motion with Constant Acceleration Maxim Tsoi Falling Objects Graphical Analysis of Linear Motion Physics Department, The University of Texas at Austin http://www.ph.utexas.edu/~tsoi/302K.htm 302K - Ch.2 302K - Ch.2 Kinematics Motion Branch of classical mechanics describes motion in terms of space and time • Ignore the agents that caused that motion continuous change in the position of an object 3 categories • Translational • Motion in one dimension (1D motion) • Concepts: position, displacement, velocity, acceleration • Rotational • Vibrational 302K - Ch.2 302K - Ch.2 Particle Model Position, Velocity, and Speed Used to describe moving objects Position • The location of the particle with respect to a chosen reference point (the origin of a coordinate system) • We describe the moving object as a particle regardless of its size • Particle is a point-like object = object with mass but having • The motion of a particle is completely known if the particle’s position in space is known at all times infinitely small size 302K - Ch.2 302K - Ch.2 1 Position, Velocity, and Speed Position, Velocity, and Speed Position Displacement • The location of the particle with respect to a chosen reference point (the origin of a coordinate system) • Particle’s change in position in some time interval Initial position • The motion of a particle is completely known if the particle’s position in space is known at all times Position-time graph xi Final position x x f x i Displacement xf Position-time graph x t 10 s x B x A 0 x t 50 s x F x A 0 302K - Ch.2 302K - Ch.2 Position, Velocity, and Speed Position, Velocity, and Speed Displacement vs Distance traveled Average velocity • Displacement Particle’s change in position in some time interval • Distance The length of a path followed by the particle • Displacement is a vector • The average velocity displacement x vx of a particle is defined as the particle’s divided by the time interval t during which that displacement occurs: vx Position-time graph quantity requires the x t L T Position-time graph • vector quantity positive specification of both or negative in 1D direction and magnitude • Interpreted geometrically • Distance scalar quantity; has a numerical by the slope of a straight value and no direction line between two points 52 m 30 m 2.2 10 s 0 302K - Ch.2 m s 302K - Ch.2 Position, Velocity, and Speed Instantaneous Velocity and Speed Speed vs Velocity How to find velocity at any moment in time? • The average speed of a particle is defined as the total distance traveled divided by the total time interval required to travel this distance: average speed total distance total time Position-time graph • Dimension L T • scalar quantity no • The instantaneous velocity vx equals the limiting value of the ratio x/t as t approaches zero: direction; no sign in 1D speed no average velocity provide information about details of the trip 302K - Ch.2 v x lim t 0 • NOTE Neither average x t instantaneous velocity is the slope of the line tangent to the position-time graph • Instantaneous velocity can be positive, negative, or zero • Instantaneous speed is the magnitude of its (instantaneous) velocity 302K - Ch.2 Example 2.2 2 Acceleration Acceleration Quantify changes in velocity Average acceleration • Velocity of a particle changes with time particle is said to be accelerating Initial velocity Final velocity v xi v xf at time at time ti tf v v xf v xi • The average acceleration of the particle is defined as the change in velocity divided by the time interval during which that change occurs: ax v x v xf v xi t t f ti • is a vector quantity t t f t i • has dimensions of L/T2 • in SI units m/s2 Velocity-time graph 302K - Ch.2 Velocity-time graph 302K - Ch.2 Acceleration Acceleration Instantaneous acceleration Direction of acceleration vs direction of velocity • The instantaneous acceleration is the limit of the average acceleration as t • When the object’s velocity and acceleration are in the same direction the object is speeding up. approaches zero: a x lim t 0 v x t instantaneous acceleration is the slope of the line tangent to the velocity-time graph • When the object’s velocity and acceleration are in opposite directions the object is slowing down Acceleration is caused by force (ch. 4): F a • Force and acceleration are both vectors acting in the same direction • Equate the direction of the acceleration to the direction of a force! Velocity-time graph 302K - Ch.2 302K - Ch.2 Finding displacement Motion Diagrams Help to describe the velocity and acceleration while an object is in motion from velocity-time graph Stroboscopic photograph of a moving car (several images of the car, taken as the strobe light flashes at a constant rate) Displacement = area under the velocity-time graph xn v xn t n x v xn t n n lim t n 0 xn t n n Velocity-time graph Describe the motion of the car in each diagram 302K - Ch.2 v 302K - Ch.2 3 Finding displacement 1D Motion with Constant Acceleration from velocity-time graph simple type of 1D motion Displacement = area under the velocity-time graph a x const • over any time interval: Examples where particle moves (a) at constant velocity (b) with a velocity that is proportional to t x v xi t x 1 2 2 t A a x t A 1 a x t A 2 ax v xf v xi t f ti ax v xf v xi t 0 a x ax v xf v xi a x t • v ~ t, i.e., we can express the average velocity in any time interval as the arithmetic mean of the initial and final velocities: vx v xi v xf 2 x t x f xi v x t 1 2 v xi v xf t • or via acceleration: x f x i 1 v xi v xi a x t t x i v xi t 1 a x t 2 2 2 302K - Ch.2 302K - Ch.2 1D Motion with Constant Acceleration 1D Motion with Constant Acceleration simple type of 1D motion Kinematic equations for 1D motion • Velocity as a function of time: • finally, expression for the final velocity that does not contain time: x f xi 1 2 v xi v xf t 1 2 v xi v xf v xf v xi ax v xf v xi a x t 2 2 v xf v xi • Position as a function of velocity and time: 2a x x f x i 1 v xi v xf t 2 2 v xf v 2 2a x x f x i xi • Position as a function of time: • Note: for zero acceleration (a=0): x f x i v xi t 1 a x t 2 2 v xf v xi v x x f xi v x t • Velocity as a function of position: 2 2 v xf v xi 2a x x f x i 302K - Ch.2 Example 2.7-8 302K - Ch.2 Freely Falling Objects SUMMARY Kinematics in One Dimension any object moving freely under the influence of gravity alone • Displacement: • Galileo Galilei (1564-1642) originated our present-day ideas x x f x i vx • Average velocity: concerning falling objects average speed • Average speed: • In the absence of air resistance, all objects dropped near the Earth’s x t total distance total time v x lim x t surface fall toward the Earth with the same constant acceleration • Instantaneous velocity: under the influence of the Earth’s gravity • Instantaneous speed of a particle is equal to the magnitude of its inst. velocity • NOTE: objects thrown upward or downward and released from rest are all falling freely once they are released t 0 • Average acceleration: ax v x v xf v xi t t f ti • Instantaneous acceleration • The magnitude of the free-fall acceleration is g=9.80 m/s2 • Equations of kinematics: v xf v xi a x t • Decreases with increasing altitude • Slightly varies with changes in latitude 2 2 v xf v xi 2a x x f x i • 1D Kinematic equations can be applied (ay=-g) a x lim t 0 v x t x f x i 1 v xi v xf t 2 x f x i v xi t 1 a x t 2 2 • An object falling freely in the presence of the Earth’s gravity experiences a free-fall acceleration directed toward the center of the Earth g=9.80 m/s2 302K - Ch.2 Example 2.10 302K - Ch.2 4 ...
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This note was uploaded on 09/01/2009 for the course PHY M taught by Professor Staff during the Spring '09 term at University of Texas at Austin.

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