A rock with mass m slides with initial velocity v_0 on a horizontal surface. A r

Question

A rock with mass m slides with initial velocity v_0 on a horizontal surface. A retarding force F_r that the surface exerts on the rock is proportional to the square root of the instantaneous velocity of the rock (F_r = - kv^(1/2)). a) Find expression for the velocity of the rock as a function of time. Express your answer in terms of the variables v_0, k , t , and m. v=? b) Find expression for the position of the rock as a function of time. Express your answer in terms of the variables v_0, k, t, and m. x= ? c) In terms of m, k, and v_0, at what time will the rock come to rest? t= ? d) In terms of m, k, and v_0, what is the distance of the rock from its starting point when it comes to rest? d=?

Explanation / Answer

a) Fr = -kv^0.5

mdv/dt = -kV^0.5

integrate[dV/V^0.5] = integrate[-k/mdt]

2 V^0.5 = -kt + C , where C = integration constant

given at t =0 , V =Vo , hence C =2Vo^0.5

V^0.5 = -kt/2 + Vo^0.5

V = [-kt/2 + Vo^0.5]^2

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