Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Falling in a viscous medium

When a small object moves in a viscous medium it experiences a resistive force that is opposite in direction to its velocity and directly proportional to its speed. For this problem, assume that the object only moves up and down and take the upward direction to be positive. We write the viscous drag force as Fv = -bv. In this problem, you will solve the differential equation for the motion in one (vertical) dimension of a body subject to gravity and viscous drag.

1. What are the dimensions of the viscous drag coefficient, b?
2. Write Newton's Second law of motion for a body falling under the influence of gravity and viscous drag.
3. From the parameters of the equation (m, b, and g) create a natural mass scale, MN, a natural length scale, LN, and a natural time scale, TN.
4. From your natural length and time scales, create a natural velocity scale, VN = LN/TN. Interpret what this velocity means physically.
5. Write Newton's second law as a differential equation for the velocity, v, and re-express it as an equation for a dimensionless velocity, V = v/VN as a function of a dimensionless time, T = t/TN. If you simplify the resulting equation, all the parameters of the problem should drop out and each of the terms should be dimensionless.
6. Solve this equation using standard techniques for first order equations.
7. If the particular solution to your first-order differential equation satisfies the initial condition, V(T=0) = v0/VN (that is, the initial velocity is v0) eliminate your integration constant in favor of v0. Rewrite your solution in terms of dimensioned variables and group your terms in a way that you can clearly explain what your solution is telling you about how the motion of the objects evolves. Give a clear and quantitative (in symbols, not in numbers!) description of what the object does as a function of time.