6. Suppose you drop a golf ball onto a hard surface from a height h. The collision with the ground causes the ball to lose energy and so it will not bounce back to its original height. The ball will then fall again to the ground, bounce back up, and continue. Assume that at each bounce the ball rises back to a height 3 of the height from which it dropped. Let hn be the height of the ball on the nth bounce, with ho = h. In this exercise we will determine the distance traveled by the ball and the time it takes to travel that distance. a. Determine a formula for h₁ in terms of h. b. Determine a formula for h₂ in terms of h. c. Determine a formula for h3 in terms of h. d. Determine a formula for hn in terms of h. e. Write an infinite series that represents the total distance traveled by the ball. Then determine the sum of this series. f. Next, let's determine the total amount of time the ball is in the air. i) When the ball is dropped from a height H, if we assume the only force acting on it is the acceleration due to gravity, then the height of the ball at time t is given by H 1 291². Use this formula to determine the time it takes for the ball to hit the ground after being dropped from height H. ii) Use your work in the preceding item, along with that in (a)-(e) above to de- termine the total amount of time the ball is in the air.
6. Suppose you drop a golf ball onto a hard surface from a height h. The collision with the ground causes the ball to lose energy and so it will not bounce back to its original height. The ball will then fall again to the ground, bounce back up, and continue. Assume that at each bounce the ball rises back to a height 3 of the height from which it dropped. Let hn be the height of the ball on the nth bounce, with ho = h. In this exercise we will determine the distance traveled by the ball and the time it takes to travel that distance. a. Determine a formula for h₁ in terms of h. b. Determine a formula for h₂ in terms of h. c. Determine a formula for h3 in terms of h. d. Determine a formula for hn in terms of h. e. Write an infinite series that represents the total distance traveled by the ball. Then determine the sum of this series. f. Next, let's determine the total amount of time the ball is in the air. i) When the ball is dropped from a height H, if we assume the only force acting on it is the acceleration due to gravity, then the height of the ball at time t is given by H 1 291². Use this formula to determine the time it takes for the ball to hit the ground after being dropped from height H. ii) Use your work in the preceding item, along with that in (a)-(e) above to de- termine the total amount of time the ball is in the air.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:6.
Suppose you drop a golf ball onto a hard surface from a height h. The collision with the
ground causes the ball to lose energy and so it will not bounce back to its original height.
The ball will then fall again to the ground, bounce back up, and continue. Assume that
at each bounce the ball rises back to a height of the height from which it dropped.
Let hn be the height of the ball on the nth bounce, with ho = h. In this exercise we will
determine the distance traveled by the ball and the time it takes to travel that distance.
a. Determine a formula for h₁ in terms of h.
b. Determine a formula for h₂ in terms of h.
c. Determine a formula for h3 in terms of h.
d. Determine a formula for hn in terms of h.
e. Write an infinite series that represents the total distance traveled by the ball. Then
determine the sum of this series.
f. Next, let's determine the total amount of time the ball is in the air.
i) When the ball is dropped from a height H, if we assume the only force acting
on it is the acceleration due to gravity, then the height of the ball at time t is
given by
1-1/2gt².
Use this formula to determine the time it takes for the ball to hit the ground
after being dropped from height H.
H
ii) Use
your work in the preceding item, along with that in (a)-(e) above to de-
termine the total amount of time the ball is in the air.
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