Kinmatic question involving differential equation

[gaobo9109]

Homework Statement

A particle of mass m falls fom rest; the resistance of the air when the speed is v is kv² where k is a constant. If s is the distance fallen in time t, prove that s = V²/2g ln(V²/(V²-v²)), where V²=mg/k

Homework Equations

The Attempt at a Solution

I have already proven that t = V/2g ln((V+v)/(V-v)). How do I make use of that equation to solve the problem? Do I have to change to v(t) equation and integrate it?

 

[ehild]

Yes, this is one way. Integrating v(t) you get s(t), but you need the s(v) relation, so you have to substitute t(v) for t.

The other way is that you transform the original differential equation:

dv/dt= g-k/m * v^2

by considering s the independent variable and applying the chain rule during differentiation with respect to t.

dv/dt=dv/ds*ds/dt = dv/ds *v .

The new differential equation is : v dv/ds = g-k/m*v^2.
This is separable, easy to solve with the condition that v=0 at s=0.



ehild