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## Homework Statement

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.)

## Homework Equations

Kinetic Energy = 1/2 mv^2

momentum = mv

work = f*d

f=ma

## 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.