Help with final velocity of a rocket

[PhysicsLoop]

Homework Statement

Calculate the final velocity of a rocket if the velocity of the ejected gas is c, initial mass of rocket is M₀ and final mass is M_r, v₀=0.

Homework Equations

We have to calculate the final velocity of rocket.

The Attempt at a Solution

Because the total momentum of an isolated system is conserved, the total system - rocket + exhausted gas - must still have zero momentum.
If the velocity of gas is c, then:
M₀*v - (M₀-m)*c=0
m - mass of gas ejected

What to do next?

 

[tiny-tim]

Welcome to PF!

Hi PhysicsLoop! Welcome to PF!

Because the total momentum of an isolated system is conserved, the total system - rocket + exhausted gas - must still have zero momentum.

Ah, but you don't know what the final velocity of the gas is (it changes according to the velocity of the rocket at the time).

You'll need to write a differential equation for what happens when the speed is v and the mass is m, and a small amount of mass dm is ejected.

 

[PhysicsLoop]

Thanks! :)
Velocity of the gas is constant equal with c (speed light), it doesn't change with rocket velocity.
Can you help me? I'm stucked.

 

[tiny-tim]

v is a function of m

write the equation for conservation of momentum when the mass changes from m to m - dm

 

[PhysicsLoop]

I still don't get it... Can you write it please?

 

[PhysicsLoop]

Can someone help me with this?