A rocket ejects exhaust with an velocity of u. The rate at which the exhaust mass is used (mass per unit time) is b. We assume that the rocket accelerates in a straight line starting initial mass (fuel plus the body and payload) be mi, and mf be its final mass, after all the fuel is used up. (a) Find the rocket's final velocity, v, in terms of u, mi, mf. (b) A typical exhaust velocity for chemical rocket engines is 4000m/s. Estimate the initial mass of a rocket that could accelerate a one-ton payload to 10% of the speed of light, and show that this design won't work. (ignore the mass of the fuel tanks.)
Kinetic Energy = 1/2 mv^2
momentum = mv
work = f*d
The Attempt at a Solution
I started out with the conservation of energy, and then switched to the conservation of momentum.
The rocket starts out at rest. so mi*vi = 0
0 = mf*v +(mi-mf)*u
which makes sense since the rocket's momentum will be opposite to the exhaust velocity, and the two momentum will cancel.
Now is the part where I am stuck.