# Advanced mechanics - x(t) from v(x)

1. Sep 25, 2011

### tourjete

1. The problem statement, all variables and given/known data
A particle of mass m's velocity varies according to bx-n

Find the position as a function of time, setting x = x0 at t=0

2. Relevant equations

v(x) = bx-n

possibly relevant: f(x) = -b2mnx-2n-1

3. The attempt at a solution

The first part of the question asked me to find the force acting on the particle as a function of x, which I did using the chain rule. I'm a little unclear as to whether I need f(x) to get x(t).

Anyway, here's my attempt at a solution:
dv/dt = (dv/dx)*(dx/dt)
dv/dt = (dv/dx) * v(x)

Both of these quantities are known so I plugged them in and got an expression for dv/dt. I then tried to integrate that expression twice, once to get v(t) and another time to get x(t). However, when I do that I just get the expression times t2/2, which would make x(0) = 0, not x0 as the problem statement gives.

Am I doing this the complete wrong way or am I on the right track and just not understanding calculus?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 25, 2011

### Dickfore

$$v = \frac{d x}{d t} = b x^{-n}$$
This is a 1st order ODE with separable variables. The variables separate as:
$$x^{n} \, dx = b \, dt$$
which can be integrated by elementary tabular integrals. The one arbitrary constant is found from the initial condition.

3. Sep 25, 2011

### tourjete

Thanks!

I solved the ODE by integrating both sides and got xn+1/(n+1) = bt. I don't see where the constant comes into play here.

4. Sep 25, 2011

### rude man

Finish the expression for the integration of your ode. What must one add to every indefinite integral of df/dx?

5. Sep 25, 2011

### tourjete

so I have:

xn+1/(n+1) = bt + C

then I plug in the initial condition x=x0 at t = 0

C = x0n+1/(n+1)

I just need to solve this for x now, correct?

6. Sep 25, 2011

### Dickfore

yes.

7. Sep 25, 2011

### rude man

Molto bene, molto bene!