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Ball

Equations

h_n = (C_R^2)^n h_0 | d_t = (h_0 (1 + C_R^2) (1 - (C_R^2)^n))/(1 - C_R^2)
d_b = (2 h_0 (1 - (C_R^2)^n))/(1 - C_R^2) - h_0 | | 
h_n | n^th bounce height
h_0 | drop height
n | bounces
C_R | restitution coefficient
d_t | total distance covered at the top of bounce n
d_b | total distance covered at the start of bounce n

Input values

drop height | 2 meters
bounces | 2
restitution coefficient | 0.5

Results

n^th bounce height | 12.5 cm (centimeters)
= 4.921 inches
= 0.4101 feet
total distance covered at the start of bounce n | 3 meters
= 9.843 feet
= 9' 10.11"
total distance covered at the top of bounce n | 3.125 meters
= 10.25 feet
= 10' 3.031"

Further results

total distance covered after infinite bounces | 3.333 meters
= 10.94 feet
= 333.3 cm (centimeters)

Bounce height derivation

C_R = | sqrt(h_1/h_0)
h_1 = | C_R^2 h_0
h_2 = | C_R^2 h_1 = C_R^4 h_0
... | 
h_n = | (C_R^2)^n h_0

Distance traveled derivations

d_b = | h_0 + sum_(m=1)^(n - 1) 2 h_m
 = | sum_(m=0)^(n - 1) 2 h_m - h_0
 = | sum_(m=0)^(n - 1) 2 (C_R^2)^m h_0 - h_0
 = | (2 h_0 (1 - (C_R^2)^n))/(1 - C_R^2) - h_0
d_t = | h_0 + sum_(m=1)^(n - 1) 2 h_m + h_n
 = | sum_(m=0)^(n - 1) 2 h_m - h_0 + h_n
 = | sum_(m=0)^(n - 1) 2 (C_R^2)^m h_0 - h_0 + h_n
 = | (2 h_0 (1 - (C_R^2)^n))/(1 - C_R^2) - h_0 + (C_R^2)^n h_0
 = | h_0 ×((2 (1 - (C_R^2)^n))/(1 - C_R^2) - 1 + (C_R^2)^n)
 = | h_0/(1 - C_R^2) ×(2 (1 - (C_R^2)^n) - (1 - C_R^2) + (1 - C_R^2) (C_R^2)^n)
 = | h_0/(1 - C_R^2) ×(1 - (C_R^2)^n + C_R^2 - C_R^2 (C_R^2)^n)
 = | (h_0 (1 + C_R^2) (1 - (C_R^2)^n))/(1 - C_R^2)

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