Reference no: EM13966716
Suppose we have an elastic ball. We drop it from a height of 256 feet falling at the normal rate of gravitational acceleration (32 feet per second per second). Assume that the bounce takes a negligible amount of time and that after the bounce the ball travels upward with a return velocity of 75% of its impact velocity.
1. The equation for the falling ball up until the first bounce is: h(t) = 256 - 16t2
By solving for height of zero we can see that the ball falls 256 feet in 4 seconds.
The equation for the ball between the first bounce and second bounce realigning time to t = 0 at the initial time of the bounce is:
h(t)=96t-16t2
This has a vertex at t=3s at a height of 144 feet. When the ball hits the ground itAc€?cs velocity is 96 feet per second.
(Use series and sequences)
a. Give the equation for the height of the ball between the second bounce and third bounce? h(t)=__t-16t2
b. What distance does the ball travel up and down on the bounce?
c. How much time does it take?
d. Give the equation for the height of the ball between the third bounce and fourth bounce? Ac„A1(t)=__t-16t2
e. What distance does the ball travel up and down on the bounce?
f. How much time does it take?
2. Give the equation for the height of the ball for the nth bounce: h(t)=___t-16t2
3. Compare the distance travelled by the ball between the first and second bounce and between the second and third bounce. Now, compare the distance travelled by the ball between the second and third bounce and between the third and fourth bounce. You should notice a geometric series pattern. Give the sigma notation for the finite geometric series for the distance the ball travels at the time of the nth bounce (if one exists).
4. Compare the time taken by the ball between the first and second bounce and between the second and third bounce. Now, compare the time taken by the ball between the second and third bounce and between the third and fourth bounce. Again a notice a geometric series pattern? Give the sigma notation finite geometric series for the time elapsed at the time of the nth bounce (if one exists).
5. How many bounces does it take for the ball to travel a total distance over 800 feet?
6.
a. How far has the ball traveled through the bounce previous to the one in question 5?
b. How much time has elapsed by then?
7. Give the equation for the height of the ball for the bounce where it goes past 800 feet.
8. Use this equation to find the time during this bounce when the ball reaches the 800 foot mark, then add it to 6b. to get the total time it takes for the ball to travel 800 feet. Round to the nearest hundredth of a second.
9. What is the total distance up and down that the ball can travel in our model (the infinite sum)?
10. What is the total time that the ball can be bouncing (the infinite sum)?
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