Calculus 1 – Spring 2008 Dr. Hamblin Section 2.5: Follow the Bouncing Ball When an object is freely falling, physics tells us that it has constant downward acceleration. If is the function that describes the height of the ball after t seconds, then the acceleration is given by or . For this example, we’ll use feet instead of meters. Using antiderivatives, we can now find formulas for h and h: The values of and depend on the initial position and initial velocity of the object. We will use these formulas to analyze a ball being dropped off of a balcony. When the ball bounces, it loses some of its energy, and rises at a slower speed than it fell. We want to measure this effect, so first we need to collect some data. We will drop a ball off of the balcony, and measure the time it takes to hit the ground the first time and the second time. Trial # 1 2 3 4 Average First bounce Second bounce Let be the average time until the first bounce, and let be the average time until the second bounce. We want to use the data for and to compute the unknown constants and above. However, when the ball bounces, its motion changes, so we will have to compute and once for the function describing the motion of the ball before the first bounce, and again for the function describing the motion of the ball between bounces. So is a piecewise function, as shown below. Calculus 1 – Spring 2008 Dr. Hamblin Before the First Bounce We would like to find the values of and so that we can understand the first “piece” of our function. We know the following two facts: . Because the ball was dropped instead of thrown, its initial velocity is 0. Because the ball hit the ground at time , we know that the height of the ball at that time was 0. Questions 1. Plug this information into our equations for and to find the values of and . 2. How high up is the balcony? (Hint: Plug the appropriate value into one of our functions.) 3. How fast was the ball traveling when it hit the ground? (Again, plug an appropriate value into one of our functions.) Between the Bounces Now we want to analyze the second “piece” of our function. This time we have the following information: . Since the ball was on the ground at time , the height function is equal to zero at this time. The ball hit the ground for the second time at time , so the height function equals zero here also. Questions 4. Use this information to find the values of and . (You will have two equations and two unknowns.) 5. How fast was the ball rising in the instant after the first bounce? 6. Using your answers to questions 3 and 5, what percentage of speed did the ball lose when it hit the ground? Moving Forward Your answer to #6 gives a measure of the elasticity of the ball. A racquetball is much more elastic than, say, a basketball. The racquet ball loses a much smaller percentage of its speed when it hits the ground, and so would bounce higher than a basketball dropped from the same height. Using this value, we can predict that after the second bounce, the ball loses the same percentage of its speed. This tells us about the third “piece” of our function, specifically what is. We also know , and this allows us to compute and . We could then predict the time when the ball would hit the ground a third time, and so on. .