# Homework Help: Variable mass dust particle

1. Apr 19, 2014

### Matt_993

1. The problem statement, all variables and given/known data

At time t=0 a dust particle of mass m_0 starts to fall from rest through a cloud. Its mass grows exponentially with the distance fallen, so that after falling through a distance x its mass is m_0exp[αx] where α is constant. Show that at time t the velocity of the particle is given by:

v=sqrt(g/α)tanh(t(sqrt(αg))

2. Relevant equations

Variable mass equation:

mg= mv'+vm'

3. The attempt at a solution

Using the variable mass equation i've fiddled around with it and gotten out a differential equation in terms of v, which is:

dv/dt=-αv^2+g

Which after filling in the answer im looking for is the correct equation. However solving it properly by hand seems harder as its a non-linear ODE. I was just wondering is there any other way of going about this problem or if not, can anyone give me a tip on how to solve the ODE properly.

Last edited: Apr 19, 2014
2. Apr 19, 2014

### TSny

Hello, Matt_993. Welcome to PF!

Did you mistype this? What does the symbol y stand for?

Once you get the equation written in terms of just 2 variables, you might try the technique of separation of variables.

3. Apr 19, 2014

### Matt_993

Yea the y should've been a v, I've changed it. But I can't seem to be able to separate the variables in the equation because of g, is there another way?

4. Apr 19, 2014

### TSny

Divide both sides of the equation by the entire right hand side.

5. Apr 19, 2014

### Matt_993

How very stupid of me, can't believe I missed that.

That works, thanks for the help :)