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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 pointlike 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
Positiontime graph xi Final position x x f x i Displacement xf Positiontime 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 Positiontime graph quantity requires the x
t L
T Positiontime 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
Positiontime 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 positiontime
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 Velocitytime graph
302K  Ch.2 Velocitytime 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 velocitytime 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! Velocitytime 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 velocitytime 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 velocitytime graph xn v xn t n
x v xn t n n lim t n 0 xn t n n Velocitytime 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 velocitytime graph simple type of 1D motion Displacement = area under the velocitytime 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 2.78 302K  Ch.2 Freely Falling Objects SUMMARY
Kinematics in One Dimension any object moving freely under the influence of gravity alone
• Displacement: • Galileo Galilei (15641642) originated our presentday 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 freefall 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
freefall 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.
 Spring '09
 Staff
 Physics

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