Relativistic Energy Question

[tommyball]

Hi-

I have a question regarding relativistic kinetic energy. If a spaceship is moving at a velocity relative to the Earth and then accelerates, to compute the work done by the engine/KE, should I use the given final velocity(the problem isn't entirely clear what this speed is in reference to) to compute $$\gamma$$ or the difference in velocity? The problem is as follows:

1.A space vehicle with a mass of 50,000 kg is moving directly away from the Earth at a speed of v₀, relative to the earth. It sends a radio signal back to the earth, notifying ground control that it is about to begin a rocket burn to accelerate to v_f. The radio frequency for the transmission is 300MHz, measured on the craft. The signal is received on Earth at 260 MHz.

A) What was v₀, before the burn?
B) If the speed of the vehicle after the burn is 0.4c, and no other forces act on the craft, how much work did the rocket engine do on the vehicle?

Homework Equations

To determine v₀ I used the Doppler Shift equation:

f=f₀ * sqrt[(1-(v/c)) / (1 + (v/c))]

To determine the work, I used the relativistic work/KE equation:

W = $$\Delta$$KE = mc^2($$\gamma$$ - 1)

The Attempt at a Solution

I computed v₀ = 0.142c

My question is, when computing $$\gamma$$ for the W/KE equation, do I use the given vf of 0.4c or since I am trying to determine the work done by the engine, do I use a v of 0.4c-0.142c = 0.258c since the v₀ is relative to the Earth and in the craft's reference frame the engine is accelerating it from 0?

Thank you,
Todd

 

[Jhero]

Hello Todd,

That's a great question! When calculating the relativistic kinetic energy, you should use the final velocity relative to the Earth. This is because the work done by the engine is in reference to the Earth's frame of reference, not the spaceship's. So, in this case, you would use the given vf of 0.4c to calculate the work done by the engine. The initial velocity, v0, is already factored into the Doppler shift equation you used to calculate v0. I hope this helps clarify things for you!