# Perturbation with equations of motion for air resistance

1. Oct 5, 2011

### hb1547

1. The problem statement, all variables and given/known data
"A ball is tossed upwards with speed $$V_0$$. Air resistance is $$-mkv^2$$ and there's gravity too.

Find the the time it takes the ball to reach the maximum height. Do not solve the equation of motion exactly. Use the perturbation method on the equation of motion. Solve the equation of motion without the air resistance. Then include the air resistance term, but plug in the first solution to the air resistance term. Find the leading order correction to x.

2. Relevant equations
F=ma

3. The attempt at a solution
It's unclear what is meant by 'solve' the E.O.M. I assume that meant find v(t).
Without air resistance, that was quick: $$v = -gt + v_o$$. Then I added the air resistance term (and cancelled mass):
$$a = -g -kv^2$$
Plugging in the above into that for v didn't seem to suggest anything -- in fact after integrating I got an upward parabola which doesn't seem to make sense.

I think the perturbation parameter might be $$\frac{kv_o^2}{g}$$ since that's unitless.

Last edited: Oct 5, 2011
2. Oct 6, 2011

### rude man

Never heard of 'perturbation parameter'. But the diff. eq. is easy to solve.

Your equation for a is wrong. g and v^2 oppose each other (unless you considered g < 0)