How Much Fuel Must a Rocket Burn to Double Its Exhaust Speed?

[themadhatter1]

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

$$m\frac{dv}{dt}=\frac{dm}{dt}\vec{v_{e}}$$
a. show that $$\vec{v}(t)=\vec{v}(0) - ln\frac{m(0)}{m(t)}\vec{v_{e}}$$

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:
$$2\vec{v_{e}}=\vec{v}(0) - ln\frac{m(0)}{m(t)}\vec{v_{e}}$$

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?

 

[process91]

It seems to be staring you in the face. You have an equation with essentially one unknown. Where in your equation is your initial mass, and what indicates the fraction of it expended at time t?

 

[themadhatter1]

Oh wait.

you know v(0)=0.

So this could be arranged to be

$$m(t)\frac{1}{e^2}=m(0)$$

so it uses 1/e² of it's fuel. Is this what you were thinking?

 

[process91]

The problem is not asking you to find m(0), the initial mass, but rather the fraction of the initial mass which the rocket would have to burn in order to reach the velocity specified. In your equation, you can treat $v_e, m(0)$, and $v(0)$ as known quantities (you are correct that v(0)=0). You should be solving in terms of the only unknown quantity, m(t). Then, the fraction of the initial mass which the rocket has burned at time t is $1-\frac{m(t)}{m(0)}$, so put your answer in that form, ie. $1-\frac{m(t)}{m(0)}=...$ and that should be it.