Calculus

posted by .

A ball is dropped from a height of 10 feet and bounces. Each bounce is ¨ú of the height of the bounce before. Thus, after the ball hits the floor for the first time, the ball rises to a height of 10(¨ú ) = 7.5 feet, and after it hits the second floor for the second time, it rises to a height of 7.5(¨ú ) = 10(¨ú )©÷ = 5.625 feet. (Assume g= 32 ft/sec ©÷ and that there is no air resistance.)
(a) Find an expression for the height to which the ball rises after it hits the floor for the nth time.
(b) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the first, second, third, and fourth times.
(c) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the nth time. Express your answer in closed-form.

For part (a) I got ar^n
For part (b) I got
1st bounce = a + ar
2nd bounce = a + ar + ar + ar©÷
3rd bounce = a + ar+ ar + ar©÷ + ar©ø
4th bounce = a + ar +ar + ar©÷ + ar©ø + ar©ø + ar©ù
For part (c) I do not kow what to do any help would be greatly appreciated.

  • Calculus -

    a) h = 10(.75)^n
    that is basically what you had

    b) d = 10 + 10(.75)^1 + 10 (.75)^2 + 10(.75)^3 .... In other words I sort of agree with you

    c) This is a geometric series like a
    compound interest problem

    in general
    g + gr + gr^2 + .... gr^(n-1)
    Sn = [ g (1-r^n ]/ (1-r)
    here g = 10 and r = .75
    so
    Sn = [ 10 (1 - .75^n) /.25

  • Calculus -

    I disagree with part b)

    at the first bounce the ball has traveled 10 ft.
    at the second bounce, it went up 7.5 and then down 7.5, so
    distance after two bounces = 10 + 2(10)(.75)^1

    after 3 bounces it went 10 + 2(10)(.75) + 2(10)(.75)^2

    so after the first bounce, you have to double the distance, since it goes up and then the same distance down again.

    so the series
    = 10 + 2(10)(.75) + 2(10)(.75)^2 + 2(10)(.75)^3 + ....
    so
    Sn = 10 + 20(.75)[1 - .75^(n-1)]/(1 - .75)

    so I see an infinite geometric series starting with the second term and that sum is
    10 + (20)(.75)/(1 - .75)
    = 10 + 15/.25
    = 70 feet

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Math

    Fay's rubber ball bounces exactly half the height from which it is dropped. She drops the ball from the top of a building that is 64 meters tall. How high will the ball bounce on its eighth bounce. Could someone explain this?
  2. math

    a ball is dropped from a height of 5m. after each bounce, the ball rises to 45% of its previous height. what the total vertical distance that the ball has travelled after it has hit the ground for for the 6th time?
  3. physics

    a superball of mass .125 kg is dropped onto a concrete floor from a height of 1m it bounces back at a height of .9m. What is the net force acting on the ball before and after its released?
  4. physics

    A ball is tossed so that it bounces off the ground, rises to a height of 0.90 m, and then hits the ground again 0.50 m away from the first bounce. How long is the ball in the air between the two bounces?
  5. physics

    A ball is tossed so that it bounces off the ground, rises to height of 2 m, and then hits the ground again 0.75 m away from the first bounce. a. How long is the ball in the air between the two bounces?
  6. physics

    A superball is dropped from rest from a height of 2.0m. It bounces repeatedly from the floor, as superballs are prone to do. After each bounce the ball dissipates some energy, so eventually it comes to rest. The following pattern is …
  7. physics

    A superball is dropped from rest from a height of 2.0m. It bounces repeatedly from the floor, as superballs are prone to do. After each bounce the ball dissipates some energy, so eventually it comes to rest. The following pattern is …
  8. 5th grade math

    On each bounce a ball reaches 4/5 of the height of its previous bounce.How high will the ball bounce on its 3rd bounce?
  9. Algebra 2

    A ball is dropped from a height of 25 feet and allowed to continue to bounce until it eventually comes to a rest. Each time the ball bounces it rebounds to 45 of its previous height. After which bounce is the ball's rebound height …
  10. algebra

    A ball is dropped from a height of 30 feet and allowed to bounce until it comes to arrest each time the ball bounces it rebounds to 3/5 of its previous high after which bounce is the boss hi 10.8 feet. I understand the height gets …

More Similar Questions