# A simple differential equation

1. May 15, 2007

### John O' Meara

A body is falling in the earth's atmosphere. The forces on it are the earth's gravitational attraction and air resistance, which is proportional to its speed v. Its acceleration a is given by a=g-kv. Expressing a in the form dv/dt, obtain an expression for v. Show that whatever its initial speed, as t becomes very large the final speed of the body tends to g/k. I seem to go wrong somewhere in this equation. And what meaning has the air resistance to this question?
$$\frac{1}{g-kv}\frac{dv}{dt}=1 \\$$ Therefore $$\int \frac{1}{g-kv} dv = \int dt \\$$ Let u = g-k*v => du/dt=-k therefore dv=-du/k. Therefore the integral = $$\frac{-1}{k}\int \frac{1}{u}du$$ = -ln(u)/k, which gives $$\frac{-\ln(g-kv)}{k}= t+c \\$$ Therefore $$\ln{(g-kv)} = -kt+kc$$ Therefore $$-kv=-g + \exp^{-kt+kc}$$ The answer is actually: $$v=\frac{g}{k}+c\exp^{-kt}$$

Last edited: May 15, 2007
2. May 15, 2007

### Dick

Divide both sides of your solution by -k. Write exp(-kt+kc)/(-k)=exp(-kt)*exp(kc)/(-k). Now define a new constant C=exp(kc)/(-k). Voila. After all, exp(kc)/(-k) is just 'some constant'.