In Example 2.7, we considered the motion of a falling ball while it was in the air. We end...

In Example 2.7, we considered the motion of a falling ball while it was in the air. We ended the example and the plot in Figure 2.26 at the moment just before the ball hit the ground. Figure Q2.2 shows the position–time graph of the ball, including a short time after it hits the ground. Use this graph to deduce the acceleration when the ball first makes contact with the ground. Explain why a is positive immediately after the ball hits the ground, while it is coming to a stop. How can a positive acceleration make the velocity become zero?

Figure Q2.2

Figure 2.26 Example 2.7. Position (y) above the ground as a function of time for a ball that falls from a bridge.

Example 2.7

Motion of a Falling Object

A ball is dropped from a bridge onto the ground below. The height of the ball above the ground as a function of time is shown in Figure 2.26. Use a graphical approach to find, as functions of time, the qualitative behavior of (a) the velocity of the ball, (b) the acceleration of the ball, and (c) the total force on the ball.

RECOGNIZE THE PRINCIPLE

Velocity is the slope of the position-time curve, so we can find v from the slope of the y–t curve in Figure 2.26. Notice that here we use y to measure the position, taking the place of x in previous examples. Acceleration is the slope of the velocity—time curve, so we can find the behavior of a once we have the behavior of the velocity. Once we have the acceleration, the total force on the ball can then be found through Newton’s second law, .

SKETCH THE PROBLEM

Figure 2.26 shows how the position of the ball varies with time, and Figure 2.27 shows a picture of the ball as it falls from the bridge. The motion of the ball is one-dimensional, falling directly downward from the bridge to the ground, and its position can be measured by its height y above the ground. This notation follows the common practice of using y to represent position along a vertical direction. Figure 2.27 also shows the y coordinate axis, with its origin at ground level.

Figure 2.27 Example 2.7. A ball is dropped from a bridge, starting from rest (v = 0) at a height h above the ground. The motion of this ball is described by the y–t graph in Figure 2.26.

IDENTIFY THE RELATIONSHIPS AND SOLVE

(a) Velocity is the change in position per unit time, so the value of v at any particular time is the slope of the y–t graph at that instant (since in this example y is our position variable). We can obtain this slope graphically by following the approach in Example 2.2 and drawing tangent lines to the y–t curve at various times in Figure 2.28A. Using estimates for the slopes of these lines leads to the qualitative velocity–time graph shown alongside the y–t curve. Note that the velocity is always negative since y decreases monotonically with time. Also, the magnitude of the velocity, which is the speed, becomes larger and larger as the ball falls. (The plot here shows v as a function of time up until just before the ball reaches the ground.)

(b) Acceleration is the slope of the v–t curve; that slope is constant in Figure 2.28A, leading to the qualitative acceleration–time graph in Figure 2.28B. The acceleration is negative because the value of v is decreasing (the value of v becomes more negative with time). The magnitude of a is approximately constant during this period.

(c) To compute the force on the ball, we use Newton’s second law. According to Equation 2.8, the total force is proportional to the acceleration. Even if we do not know the sources of all the forces on the ball, we can still compute the total force from Newton’s second law. We can rearrange Newton’s second law (Eq. 2.8) as and thus arrive at the qualitative force–time graph in Figure 2.28C.

Figure 2.28 Example 2.7. A Position as a function of time for the falling ball from Figure 2.26. The slopes of the tangent lines give the ball’s velocity at three instants in time. The velocity of the ball as a function of time is obtained from plotting these slopes. B The ball’s acceleration is constant. C The force on the ball is proportional to the ball’s acceleration.

What have we learned?

Given the behavior of the position as a function of time, we can deduce (by estimating slopes) the qualitative behavior of both the velocity and the acceleration as functions of time. The behavior of the force can then be found by using Newton’s second law. For this falling ball, the total force is negative (Fig. 2.28C), meaning that the force is directed downward, along the —y direction. This force is just the gravitational force acting on the ball.

Example 2.2

Computing Velocity Using a Graphical Method

A hypothetical object moves according to the x—t graph shown in Figure 2.11A. This object is initially (when t is near t1 moving to the right, in the “positive” x direction. The object reverses direction near t2 and t3, and it is again moving to the right at the end when t is near t4. (a) Sketch the qualitative behavior of the velocity of the object as a function of time using a graphical approach. (b) Estimate the average velocity during the interval between t1 = 1.0 s and t2 = 2.5 s.

RECOGNIZE THE PRINCIPLE

For part (a), we want to find the velocity—which means the instantaneous velocity— so we need to estimate the slope of the x—t curve as a function of time. For part (b), we use the fact that the average velocity over the interval t = 1.0 s to t = 2.5 s is the slope of the x—t curve during this interval.

SKETCH THE PROBLEM

Figure 2.11B shows the x—t graph again, this time with lines drawn tangent to the x—t curve at various instants. The slopes of these tangent lines are the velocities at particular times t1, t2,... as indicated in Figure 2.11A.

IDENTIFY THE RELATIONSHIPS AND SOLVE

(a) At t = t1 the x—t slope is large and positive, so our result for v in Figure 2.12A is large and positive at t = t1. At t = t2, the x—t slope is approximately zero and hence v is near zero. At t = t3, the object is moving toward smaller values of position x, so the slope of the x—t curve and hence also the velocity are negative. Finally, at t = t4, the object is again moving to the right as x is increasing with time, so v is again positive. After estimating the x—t slope at these places, we can construct the smooth v-t curve shown in Figure 2.12A. This figure shows the qualitative behavior of the object’s velocity as a function of time.

(b) To estimate the average velocity between t1 and t2, we refer to Figure 2.12B, which shows a line segment that connects these two points on the position-time graph. The slope of this segment is the average velocity:

Reading the values of x1, x2, t1, and t2 from that graph, we find

What have we learned?

The (instantaneous) velocity is the slope of the position-time graph. To find the qualitative behavior of the velocity, we found approximate values by drawing lines tangent to the x—t curve at several places and estimating their slopes. The value of v at a particular value of t is always equal to the slope of the x—t curve at that time.

Figure 2.11 Example 2.2. A Hypothetical position-time graph. The slopes of the tangent lines in B are equal to the velocity at various instants in time.

Figure 2.12 Example 2.2. A Qualitative plot of the velocities obtained from the slopes in Figure 2.11B. B Calculation of the average velocity during the interval between t1 = 1.0 s and t2 = 2.5 s.