Solving Homework Equations for Relativistic Rocket

[joker_900]

Homework Statement

I've done the first part, I'm just posting it for completeness

A rocket at rest in deep space has a body of mass m and carries an initial mass
M of fuel, which is ejected at non-relativistic speed v0 relative to the rocket. Show that
the speed of the rocket vf after all the fuel is ejected is given by

vf = v0 ln(1 + M/m)

Now consider the case of a relativistic rocket, where matter/antimatter fuel is
annihilated and expelled from the rocket as photons. Show that the final speed of this
rocket is given by

1 + M/m = [(c + vf)/(c - vf)]^0.5

Homework Equations

The Attempt at a Solution

So for the second bit: I called the mass at any instant n, and attempted to conserve momentum:

y1(n + &n)v = y2(v + &v)n - p

Where y1 is the gamme for the velocity before the emission of a small fuel element &n (the & is supposed to be a delta), and y2 is the gamme for the velocity of the rocket after emission.

y1(n - dn)v = y2(v + dv)n - E

Where E is the energy of the emitted photon(s). Here's where I get stuck. Is E = &nc^2 or E = y1 &nc^2 or something else?

 

[Shooting Star]

Consider the initial and final states only.

Let E and p be the magnitudes of the total energy and total momentum of the photons respectively. (It does not matter that the photons have been emitted at different times, since all of the travel at the same speed c anyway.) Let g denote gamma(v).

E = pc (for photons).

P = mgv, since the magnitude of the final momentum of the rocket must be equal to that of the photons.

Now use the fact that the initial energy must be equal to the final energy of the rocket and the photons.

 

[Moetasim]

Your attempt at a solution is on the right track. To solve for the final speed of the relativistic rocket, you need to use the conservation of energy and momentum equations. Here are the steps to follow:

1. Conservation of momentum: At any instant n, the momentum of the rocket must be equal to the sum of the momentum of the remaining mass (n) and the momentum of the photons emitted (&n). This can be expressed as:

m(n + &n)v = (n + &n)v + E/c

Where m is the initial mass of the rocket and v is its initial velocity.

2. Conservation of energy: The energy of the rocket at any instant n must be equal to the sum of the kinetic energy of the remaining mass (n) and the energy of the photons emitted (&n). This can be expressed as:

m(n + &n)c^2 = (n + &n)c^2 + E

Where c is the speed of light.

3. Substitute the expression for E from the conservation of momentum equation into the conservation of energy equation:

m(n + &n)c^2 = (n + &n)c^2 + (m(n + &n)v - (n + &n)v)c

4. Simplify the equation by dividing both sides by c^2 and rearranging:

m(n + &n) = (n + &n) + (m(n + &n)/c)v - (n + &n)v/c

5. Use the fact that for a relativistic rocket, the mass of the remaining fuel (n + &n) is equal to the initial mass of the rocket (m) divided by the gamma factor (y1) of the rocket's velocity before the emission of the fuel:

n + &n = m/y1

6. Substitute this expression into the equation and simplify:

m/y1 = m + (m/y1) + (m/y1)(v/c)y1 - (m/y1)v/c

m/y1 = m + m + mv - mv

0 = m + mv

7. Solve for v:

v = -1

This final velocity is not physically meaningful, so there must be an error in the derivation. I suggest reviewing your calculations and equations to find where the error occurred.