# Rocket propelled by a giant monochromatic laser

I am preparing for my quals and stumbled across the following problem, and although it only requires undergraduate-level physics, I feel I can't piece everything together.

"A rocket of mass $m_0$ is propelled by a giant monochromatic laser mounted on the back of the rocket. The laser emits a beam with a power of $P_0$ watts and a frequency $f_0$, both measured in the rest frame of the rocket. When the beam is turned on, the rocket is driven in the opposite direction by the recoil.

(a) At $t=0$, the laser is turned on, with the rocket's velocity initially at rest in the Earth's reference frame. Calculate the instantaneous acceleration of the rocket.

(b) If the rocket is moving at velocity $v$, what is the instantaneous beam power as measured in the Earth's reference frame?

(c) The laser is kept going until the speed of the rocket reaches $v=0.9c$. What is the rest mass of the rocket at this point?"

I work in units where $c=1$. For (a), in the Earth's reference frame, I get a momentum $p(t) = \gamma m_0 v + h f$, where $\gamma = \frac{1}{\sqrt{1-v^2}}$ and $f = f_0 \sqrt{\frac{1-v}{1+v}}$ because of the Doppler shift in frequency as the rocket travels further away. Differentiating w.r.t time, I get

$$\frac{dp}{dt} = \gamma^3 m_0 a - \frac{\gamma a}{(1+v)} h f_0$$

This is as far as I get. Equaling $dp/dt=0$ for conservation of momentum make the $a$ cancel out, and there isn't any other time dependency to differentiate $v(t)$.

For part (b), we can say that in the frame of the rocket, $P_0 = r h f_0$, where $r$ is the rate of emission of photons. In the frame of the Earth, we can therefore write

$$P(t) = \frac{dE}{dt} = \frac{d}{dt} \left( \gamma m_0 + r h f \right) = m_0 \gamma^3 v a - \frac{\gamma a}{(1+v)} P_0$$

but since I don't have $a$, there's not much I can do. Also, I am not sure if $r$ needs to be corrected with a proper time factor when we boost between frames.

Also, for part (c), my first intuition was to say that the rest mass was $m_0$, but now I am thinking that the rocket's total mass decreases because we need to account for the energy lost from the laser. I am slightly confused as to what I need to take into account.

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I think you are complicating things too much. For instance, the thrust from the photons is $P_0/c$ and therefore the acceleration asked under a) is just $P_0 \ / m_0 c$. –  Johannes Aug 25 '13 at 18:14