Consider initial motion of a particle with frictional damping that is proportional to the vector velocity with damping coefficient $(F=-rv)$ Assum an initial condition of $u = U$ and $v = V$ at $t = 0$. Derive and describe the solution for the trajectory of the particle in the absence of any forcing other than friction.
I tried looking this up and all I can find is harmonic oscillators but the question says friction only and does not specify to be on a spring.
Is there an ODE out there that can help me solve this?
We are only interested when only friction acts on the particle. Newton's second law gives,
$$F=ma=mv'=-rv$$
This means
$$\int m \frac{v'}{v} dt = \int -r dt$$
$$m\ln|v(t)|=-rt+c$$
Let $t=0$
$$m \ln|V|=c$$
So that,
$$m\ln |\frac{v(t)}{V}|=-rt$$
$$v(t)=Ve^{\frac{-rt}{m}}$$
Integrating both sides of this with respect to time then letting $t=0$,
$$u(t)=-\frac{m}{r}Ve^{\frac{-rt}{m}}+c_2$$
$$U=-\frac{m}{r}V+c_2$$
$$u(t)=(U+\frac{m}{r}V)-\frac{m}{r}Ve^{\frac{-rt}{m}}$$
This an exponential function with a horizontal asymptote,
$$U+\frac{m}{r}V$$