A small particle of mass m is released from rest

A small particle of mass m is released from rest in a viscous fluid to fall under gravity. The particle experiences a linear drag force F = -bv, measuring distance z downwards as the positive direction, with v = ż. You may assume that the buoyancy force is negligible and can be ignored.

(a) What are the dimensions of the drag coefficient b?

(b) Draw a diagram of the forces acting on the particle when it is released, and when it reaches terminal velocity.

(c) Show that the terminal velocity of the particle is given by mg/b.

(d) Rearrange the force law to get an equation of motion for the particle, i.e. an equation for the acceleration z̈.

(e) Show that the expression

is a solution for the equation of motion. This requires that you check has the correct dimensions, it satisfies the correct initial conditions (z = 0, ż = 0 when t = 0), and that, when you substitute the expression into the equation of motion, the left-hand side agrees with the right.

(f) Sketch the solution from the previous part, and the velocity ż. Can you estimate an expression for the time t when the particle is traveling close to the terminal velocity?