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## Homework Statement

A rocket burning it's onboard fuel while moving through space has a velocity

**v**(t) and mass m(t) at time t. If the exhaust gasses escape with velocity

**v**

_{e}relative to the rocket , it can be deduced from Newton's Second Law of Motion that

[tex] m\frac{dv}{dt}=\frac{dm}{dt}\vec{v_{e}}

[/tex]

a. show that [tex]\vec{v}(t)=\vec{v}(0) - ln\frac{m(0)}{m(t)}\vec{v_{e}}[/tex]

b. For the rocket to accelerate in a straight line from rest to twice the speed of it's own exhaust gasses, what fraction of initial mass would the rocket have to burn as fuel?

## Homework Equations

## The Attempt at a Solution

I already solved part a, I just can't get part b.

I know that:

[tex]2\vec{v_{e}}=\vec{v}(0) - ln\frac{m(0)}{m(t)}\vec{v_{e}}[/tex]

However, I don't know where to go from here to find the initial mass the rocket would have to burn to achieve this velocity. Any ideas?