AP Physics C FRQ: A sphere sinking through a liquid

Mathematical Routines, 10 pointsAbout 22 minutes · 10 points

A small sphere of mass m is released from rest just below the surface of a deep tank of liquid and sinks vertically. The liquid exerts a resistive force of magnitude bv on the sphere, where v is its speed and b is a positive constant. Buoyancy is negligible. Take downward as positive.

(A) Describe a free-body diagram of the sphere while it sinks at a speed less than its terminal speed. For each force, give its direction and a label, and say how the lengths of the arrows compare. 2 points

(B) (i) Starting from Newton's second law, write a differential equation for v. (ii) Solve it to derive an expression for v(t), in terms of m, b, g and t. 3 points

(C) m = 0.050 kg and b = 0.20 kg/s. (i) Calculate the sphere's terminal speed. (ii) Calculate the time after release at which its speed is 90% of the terminal speed. 3 points

(D) A second sphere of the same size, so with the same b, has twice the mass. Predict how its terminal speed compares with the first sphere's, and justify your prediction. 2 points