How long will it take a log to float from the dam to the grove?

The beavers noticed that swimming from the dam to the grove along the river takes time t₁, and swimming back along the river from the grove to the dam takes time t₂, and t₁ < t₂.

How long (t=?) will it take a log to float from the dam to the grove?



Solution



t₁ (V_s + V_r)  =  L

t₂ (V_s - V_r)  =  L

t  V_r  =  L


 (V_s + V_r)  =  L/t₁

 (V_s - V_r)  =  L/t₂

(V_s + V_r) - (V_s - V_r)   =   L/t₁ - L/t₂ 

2 V_r    =     L(1/t₁ - 1/t₂)

V_r   =    L(1/t₁ - 1/t₂)/2

t =  L/ V_r  = L /  ( L(1/t₁ - 1/t₂)/2)     =  2t₁ t₂ / (t₂ - t₁)