Help on a Physics Problem?

Question

A potato moves under earths gravity with an air resistance force of -kmv^s

find the maximum height of the potato if it is shot directly upward with a student made potato gun with initial velocity vo.

the main thing here is that i dont know how to solve it. what technique, i dont just need an answer but a little explanation to your method would be nice too. thank you

Answer 1

The total downward force acting on the potato is

mg + kmv^s
which is equal to ma ( mass * acceleration)

hence a = (mg + kmv^s) / m = g + kv^s

Also a = dv / dt ( rate of change of velocity)
Hence you have got the differential equation.
This differential equation is of first order and can be solved easily using the integrating factor. Refer to any book of mathematics.
Initial velocity will be used to get the value of the constant that you get after integrating.
Well when you have got the velocity function , integrating it will give displacement. Find the time at which velocity becomes zero and put that time in displacement function to get the maximum height.

Answer 2

Assuming you meant F=-kmv^2
a = dv/dt = (dv/ds)(ds/dt) =
vdv/ds = -(g +kv^2) = -k(g/k + v^2)
-kds = vdv/(v^2 + g/k)
-2ks = ln(v^2 + g/k)
-2k(s-s0) = ln(g/k) - ln(v0^2 + gk) = ln((g/k)/(v0^2 + g/k)
k(s-s0) = [ln((kv0^2)/g + 1)]
s - s0 = [ln((kv0^2)/g + 1)]/k
s = [ln((kv0^2)/g + 1)]/k + s0