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    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

    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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