Hi all, there's just a few questions I'm having trouble with regarding relativity and I was wondering if someone could help me out.

1. a) How much work W must be done on a particle with a mass of m to accelerate it from rest to a speed of 0.894c ?

I thought this was just: E = (1/sqrt(1- (u/c)^2))(mc^2) as the total energy (and E = mc^2 is the rest energy). (u/c)^2 is .894^2, and when I followed through on that, it turns out I was incorrect (I got something like 0.49 J).

b) How much work W must be done on a particle with a mass of m to accelerate it from a speed of 0.894c to a speed of 0.982c ?

However, I figure I should understand part a first before doing part b.

2. Consider three galaxies, Alpha, Beta and Gamma. An observer in Beta sees the other two galaxies each moving away from him in opposite directions at speed 0.70c. At what speed would an observer in Alpha see the galaxy Beta moving?

In this problem, I wanted to define my frames: s' as Alpha, s as Beta (Alpha is the moving observer).

v' = (v-u)/(1-uv/c^2)

In this case, using the Lorenz velocity transformation, I thought v = 0.7c (the velocity in frame s), u = -0.7c (velocity of s' relative to s):

v' = (v-u)/(1-(uv)/c^2)

However, obviously this has to be incorrect as v' gives something like 2.7c. Am I defining my frames incorrectly?

3. An electron has a speed of 0.837 c. Through what potential difference would the electron need to be accelerated (starting from rest) in order to reach this speed? (c = 3.00 × 10^8 m/s, e = 1.60 × 10^-19 C, m el = 9.11 × 10^-31 kg)

I thought this was similar to problem 1, but I figure I should make sure I know how to do problem one. Is my potential difference, delta V, just the difference in energy from the final speed vs. the initial energy (which would just be mc^2)?

4. Assume that a certain city consumes electrical energy at an average rate of 2.0 × 10^9 W. What would be the mass change in producing enough energy to keep this city running for 12 weeks? (c = 3.00 × 10^8 m/s)

I know Force/Area = Power. Would I just try to find something in terms of period (since I have a time number) and try to apply this? Not quite sure how to go about this one.

Any help with any of these questions would be greatly appreciated. Thank you so much for looking!