Velocity as a function of distance problem

[Slepton]

Homework Statement

The velocity of a ball of mass 'm' varies with the distance v(x) = ax^-n, where a and n are positive constants.

Determine F(x), x(t) and F(t).

Homework Equations

F = ma
a = dv/dt
v = dx/dt

The Attempt at a Solution

If velocity were the function of time, i would have done it with no problem. I determined the acceleration, which is -nax^-n-1.
For v, i integrated 'a' with respect to t. But velocity's dependence in distance confused me. Any help will be highly appreciated.

 

[aim1732]

There is a simple way for these.
a=dv/dt. This can be written as a=dx/dt*dv/dx.

 

[rl.bhat]

v = dx/dt = a*x^-n
So x^n*dx = a*dt.
Find the integration and then find x in terms of a, n and t.
Then d^2(x)/dt^2 will give acceleration in terms of a,n and t. from that you can find F(t)
Acceleration a = dv/dt = dv/dx*dx/dt = dv/dx*v. Then F(x) = ma

 

[Slepton]

To rl.bhat

Thanks. I tried to follow your instruction.
Integration of x^n*dx = a*dt gave me (1/n+1)x^(n+1) = v. I didn't find x in terms of a, n and t.

 

[rl.bhat]

(1/n+1)x^(n+1) = v.
This should be
(1/n+1)x^(n+1) = at
x^(n+1) =a(n+1)*t
x = ...?