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Answer:
Total Time = 4.51 s
Step-by-step explanation:
Solution:
- It firstly asks you to prove that that statement is true. To prove it, we will need a little bit of kinematics:
y = v_o*t + 0.5*a*t^2
Where, v_o : Initial velocity = 0 ... dropped
a: Acceleration due to gravity = 32 ft / s^2
y = h ( Initial height )
h = 0 + 0.5*32*t^2
t^2 = 2*h / 32
t = 0.25*√h ...... Proven
- We know that ball rebounds back to 7/8 of its previous height h. So we will calculate times for each bounce:
- How long it has been bouncing at nth bounce, we will look at the pattern between 1st, 2nd and 3rd and 4th bounce times calculated above. We see it follows a geometric series with formula:
Total Time ( nth bounce ) = Sum to nth ( )
- The formula for sum to infinity for geometric progression is:
S∞ = a / 1 - r
Where, a = 15 , r = ( 7 / 8 )
S∞ = 15 / 1 - (7/8) = 15 / (1/8)
S∞ = 120
- Then we have:
Total Time = 0.5*√S∞ - 0.25*√15
Total Time = 0.5*√120 - 0.25*√15
Total Time = 4.51 s