Accelerating a space ship to light speeds

[TheIsingGuy]

Homework Statement

This question basically tries to investigate the feasibility of using a rocket engine to acclerate a spaceship to relativistic speeds, as with any rocket engine fule is ejected at high velocity and spaceship accelerates to conserve momentum. only that in this situation, the exhuast speed Vex is close to the speed of light

i) Express total energy and momentum of an object of rest mass m and velocity v in terms of m,v,c and $$\gamma_{v}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

Homework Equations

The Attempt at a Solution

$$P_{T}=\gamma_{V_{ex}}mv$$

$$E_{T}=\sqrt{(mc^{2})^{2}+p^{2}c^{2}}$$

so $$E_{T}=\sqrt{(mc^{2})^{2}+(\gamma_{V_{ex}}mv)^{2}c^{2}}$$

Homework Statement

Consider the inertial frame of reference in which the spaceship is instantaneously at rest at time t. During the intercal from t to t+dt, an amount of fuel of rest mass $$dm_{f}$$ is ejected in the -x direction at the exhuast speed $$v_{ex}$$ and the spaceship accelerates from rest to velocity dv. The mass of the space hsip reduces from m to m+dm, where dm is negative. Since the spaceship starts from rest, its final speed dv is not relativistic in this frame.

Now here is where things gets problematic

i) Bearing in mind that the exhaust speed is relativistic, use the principle of conservation of energy to show that $$dm=-\gamma_{v_{ex}}dm_{f}$$. Explain why is |dm|greater than $$|dm_{f}|$$.

ii) find an expression for dv using conservation of momentum.

Homework Equations

The Attempt at a Solution

I just could not figure out how to do these 2 parts, now I know that if the total energy when the ship is at rest is equal to the rest energy, then that value should be conserved, and so the total energy when the ship is moving should also equal to that value, just that there would be two opposite KEs cancelling each other out am I right in saying this?

Any help is appreciated Thanks

 

[TheIsingGuy]

I wonder if anyone replies under 2 hours

 

[TheIsingGuy]

This question has been solved. If you want to know how i did it, pm me thnx