# 4 Acceleration of an relativistic rocket

1. May 23, 2012

### bayners123

1. The problem statement, all variables and given/known data

A rocket of (time dependant) mass M ejects fuel such that its change in mass in the instantaneous ZMF is $$\frac{dM}{d\tau} = -\frac{E}{c^2}$$ The speed of the fuel ejected is $w$.
Prove that $$a = \frac{Ew}{Mc^2}$$
where a is defined by $-a^2 = A_\mu A^\mu$

3. The attempt at a solution

In the rest frame, $$P_{before} = \left(\begin{array}{c} M \\ 0 \end{array}\right) = P_{rocket} + P_{fuel} = P_{rocket} + \left(\begin{array}{c} \delta m \gamma \\ -\delta m \gamma w \end{array}\right)$$

We are given $$\frac{dM}{d\tau} = -E$$ so $$\delta m = -\frac{dM}{d\tau} d\tau = E d\tau$$

$$P \equiv M U \\ \frac{dP}{d\tau} = \frac{dM}{d\tau}U + MA = \left(\begin{array}{c} \frac{dM}{d\tau} - \gamma E \\ \gamma E w \end{array}\right) \\ -EU + MA = \left(\begin{array}{c} -E - \gamma E \\ \gamma E w \end{array}\right)$$

The rocket starts stationary in the ZMF so $U = \left(\begin{array}{c} 1 \\ 0 \end{array}\right)$

Therefore, $$A = -\frac{E}{M} \left(\begin{array}{c} \gamma \\ -\gamma w \end{array}\right)$$

$$-a^2 \equiv A_\mu A^\mu = \left(\frac{E}{M}\right)^2 [\gamma^2(w^2-1)] = \left(\frac{E}{M}\right)^2 \frac{w^2-1}{1 - w^2} = -\left(\frac{E}{M}\right)^2$$

I've lost an $w^2$ somewhere! Can anyone see it?