Suppose that a pearl is sinking in a thick fluid, like shampoo, subject to a frictional force opposing its fall and proportional to its velocity. Suppose that there is also a resistive buoyant force exerted by the shampoo. According to Archimedes’ principle, the buoyant force equals the weight of the fluid displaced by the pearl. Using m for the mass of the pearl and P for the mass of the shampoo displaced by the pearl as it descends, complete the following steps.

a. Draw a schematic diagram showing the forces acting on the pearl as it sinks, as in Figure 9.19.

b. Using ν(t) for the pearl’s velocity as a function of time t, write a differential equation modeling the velocity of the pearl as a falling body.

c. Construct a phase line displaying the signs of ν′ and ν″.

d. Sketch typical solution curves.

e. What is the terminal velocity of the pearl?

F = kv m y positive - y = 0 Fp = mg FIGURE 9.19 An object falling under the propulsion due to gravity, with a resistive force assumed to be proportional to the velocity.