Question

Captin Kirk has been forced to take the Enterprise to the center of the galaxy, which is approximately 26000 light-years from the Earth, but the warp drive is broken so that the ship will be limited to speeds loss than that of light.

1. what force would it take to accelerate such a mass at 1.0g(9.8m/s^2) to its final velocity , and how long would this take , starting from v=0?

2. what would the wavelength, frequency, and color of gamma-ray photons emitted by the ship headed back towards the Earth look like on Earth if emitted at a wavelength corresponding to the highest ever observed in real life for gamma rays in natural radiation?

3. If the Enterprise slows down to o.4c and fires torpedoes at 0.5c at a Klingon vessel approaching it at 0.3c then how fast are those torpedoes approaching the Klingons, from their perspective?

Answer 1

SOLUTION:

rest mass of the ship mo = 200e+6 kg

effective mass me = mo* [\gamma] = 2.0e+8 *520 = 1.04e+9 kg

1) final vel v = 0.9999985c

   accele   a = 1g

Force = ma =

a = (F/m) (1-v2/c2)3/2

F = a* m / (1-v2/c2)3/2 = 9.8*2.0e+8*9.8 /(1-0.9999985)3/2 = 1.067 e+18 N

as the speed increases m increases and the force has to increase to give the same acceleration

3) First we have to find the vel. of killigon reletive to our ship Enterprise.

relative to earth speed of enterprise v = 0.4c

                                         killingon     ux = 0.3c

now we can consider killigon as an object moving to the left with speed ux = -0.3c

using lorentz transvel. transform

ux' = (-0.3c -0.4c)/ (1+0.4c*0.3c/c2) = -0.625c

reletive to Enterprise the torpedo has a speed og 0.5c and the killingon has a speed of -0.625c

Now we again use Lorentz transform to find the speed of the torpedo releative to the killingon

ux" = 0.5c+0.625c/( 1+ 0.5c*0.625c/c2) = 0.857c