A small particle of weight W falling through a quescent (still) fluid experience

Question

A small particle of weight W falling through a quescent (still) fluid experiences a vertical force of F that resists the particle's motion through the fluid. For a small particle moving slowly though a fuid, the magnitude of this force is proportional to the speed of the particle C, or F = kV, where V is the particle speed and k is a constant. Draw a free body diagram of the forces along the vertical acting on this particle. Use this diagram to determine the time T required for the particle to accelerate from rest to 95% of its terminal speed, V_t, in terms of k, W, and g. Then find the distance Y traveled by the particle while it accelerates to a speed of C = .95V_t.

I have worked for an hour on this problem and can't get anywhere. Please help.

Explanation / Answer

Assume the mass of the particle to be m kg.

The particle starts from rest. When V = 0, so F = k*0 = 0, thus no resistive force acts and the only force acting at rest is W downwards. As particle moves downwards, V increases, due to which F increases. When F becomes equal to W, no force acts and body keeps moving at the velocity attained at that time, called the terminal velocity Vt. During the course when the speed increases from 0 to Vt,a net force = W-F acts downwards. This Fnet is not constant, because F is not constant.

Here we have taken 0.95Vt = V

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