A stone, of mass m, falls vertically downwards under gravity through still water. At time t, the stone has speed v and it experiences a resistance force of magnitude lmv, where l is a constant.

A stone, of mass m, falls vertically downwards under gravity through still water. At time t, the stone has speed v and it experiences a resistance force of magnitude lmv, where l is a constant.  QUESTION: a. Show dv/dt = g - lv b. If initiail speed of stone is u, find an an expression for v at time, t. ANSWER a. F = ma, and a = dv/dt. So m*dv/dt = mg - mlv. Therefore, dv/dt = g - lv b. On integration, -1/l ln (g-lv) = t + c, Substituting in the boundary conditions, the integration constant is found to be c = -1/l ln(g - lu) So ln (g - lv) = -lt + ln (g-lu) (g - lv)/(g - lu) = e^ -lt g - lv = (g - lu)e^ - lt v = 1/l (g - (g - lu)e^ -lt)