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Chapter 2. Motion Along a Straight Line. Goals for Chapter 2. To describe straight-line motion in terms of velocity and acceleration To distinguish between average and instantaneous velocity and average and instantaneous acceleration
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Chapter 2 Motion Along a Straight Line
Goals for Chapter 2 • To describe straight-line motion in terms of velocity and acceleration • To distinguish between average and instantaneous velocity and average and instantaneous acceleration • To interpret graphs of position versus time, velocity versus time, and acceleration versus time for straight-line motion • To understand straight-line motion with constant acceleration • To examine freely falling bodies • To analyze straight-line motion when the acceleration is not constant
Introduction • Kinematics is the study of motion. • Velocity and acceleration are important physical quantities. • A bungee jumper speeds up during the first part of his fall and then slows to a halt.
Displacement, time, and average velocity—Figure 2.1 • A particle moving along the x-axis has a coordinate x. • The change in the particle’s coordinate is x = x2 x1. • The average x-velocity of the particle is vav-x = x/t. • Figure 2.1 illustrates how these quantities are related.
Negative velocity • The average x-velocity is negative during a time interval if the particle moves in the negative x-direction for that time interval. Figure 2.2 illustrates this situation.
A position-time graph—Figure 2.3 • A position-time graph (an x-t graph) shows the particle’s position x as a function of time t. • Figure 2.3 shows how the average x-velocity is related to the slope of an x-t graph.
Instantaneous velocity—Figure 2.4 • The instantaneous velocity is the velocity at a specific instant of time or specific point along the path and is given by vx = dx/dt. • The average speed is not the magnitude of the average velocity!
Average and instantaneous velocities Example : A cheetah is crouched 20 m to the east of an observer’s vehicle. At time t=0 the cheetah charges an antelope and begins to run along a straight line.
Finding velocity on an x-t graph • At any point on an x-t graph, the instantaneous x-velocity is equal to the slope of the tangent to the curve at that point.
Motion diagrams • A motion diagram shows the position of a particle at various instants, and arrows represent its velocity at each instant. • Figure 2.8 shows the x-t graph and the motion diagram for a moving particle.
Average acceleration • Acceleration describes the rate of change of velocity with time. • The average x-acceleration is aav-x= vx/t. • Follow Example 2.2 for an astronaut.
Instantaneous acceleration • The instantaneous accelerationis ax = dvx/dt. • Follow Example 2.3, which illustrates an accelerating racing car.
Findingacceleration on a vx-t graph • As shown in Figure 2.12, the x-t graph may be used to find the instantaneous acceleration and the average acceleration.
A vx-t graph and a motion diagram • Figure 2.13 shows the vx-t graph and the motion diagram for a particle.
An x-t graph and a motion diagram • Figure 2.14 shows the x-t graph and the motion diagram for a particle.
Motion with constant acceleration—Figures 2.15 and 2.17 • For a particle with constant acceleration, the velocity changes at the same rate throughout the motion.
The equations of motion with constant acceleration • The four equations shown to the right apply to any straight-line motion with constant acceleration ax. • Follow the steps in Problem-Solving Strategy 2.1.
A motorcycle with constant acceleration • Follow Example 2.4 for an accelerating motorcycle.
Freely falling bodies • Free fall is the motion of an object under the influence of only gravity. • In the figure, a strobe light flashes with equal time intervals between flashes. • The velocity change is the same in each time interval, so the acceleration is constant.
A freely falling coin • Aristotle thought that heavy bodies fall faster than light ones, but Galileo showed that all bodies fall at the same rate. • If there is no air resistance, the downward acceleration of any freely falling object is g = 9.8 m/s2 = 32 ft/s2. • Follow Example 2.6 for a coin dropped from the Leaning Tower of Pisa.
Up-and-down motion in free fall • You throw a ball vertically upward from the roof of a tall building. The ball leaves your hand at a point even with the roof railing with an upward speed of 15.0 m/s. On the way down it just misses the railing.
Up-and-down motion in free fall • Find: • The position and velocity of the ball 1.00 s and 4.00 s after leaving your hand; • The velocity when the ball is 5.00 m above the railing; • The maximum height reached and the time at which it is reached; and • The acceleration of the ball when it is at its maximum height.
Is the acceleration zero at the highest point?—Figure 2.25 • The vertical velocity, but not the acceleration, is zero at the highest point.
Two solutions or one? • We return to the ball in the previous example. • How many solutions make physical sense? • Follow Example 2.8.
Velocity and position by integration • The acceleration of a car is not always constant. • The motion may be integrated over many small time intervals to give
Motion with changing acceleration • Follow Example 2.9. • Figure 2.29 illustrates the motion graphically.
