Motion Along a Straight Line Chapter 2 Motion Along a Straight Line Modifications by Mike Brotherton

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 In Example 2.1, the cheetah’s instantaneous velocity increases with time. (Follow Example 2.1)

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 vs. instantaneous acceleration Acceleration describes the rate of change of velocity with time. The average x-acceleration is aav-x = vx/t. The instantaneous acceleration is ax = dvx/dt.

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 Moving Man Simulation There is a link to a simulation on the course webpage. If you have any concerns or confusion about these plots of motion vs. time, or just want more practice, please play around with it (learning can be fun!).

The equations of motion with constant acceleration The four equations shown to the right apply to any straight-line motion with constant acceleration ax. We can work this out with a little integral calculus, but it’s also possible to do with just algebra for this specific case.

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. Page 53 in text

Up-and-down motion in free fall An object is in free fall even when it is moving upward. Instead of a ball, let’s pretend it’s something more interesting like a blood-filled pumpkin or a human heart. Page 54 in txt.

Is the acceleration zero at the highest point?—Figure 2.25 The vertical velocity, but not the acceleration, is zero at the highest point. Think about our hammer vs. feather moon video. How could you use the information there to calculate the acceleration due to the moon’s surface gravity?

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