Answer
The length of the spaceship as measured by astronauts on the space station is $~~17.65~m$
Work Step by Step
We can find the energy generated by the fuel:
$E = mc^2$
$E = (2000~kg)(3.0\times 10^8~m/s)^2$
$E = 1.8\times 10^{20}~J$
We can set the spaceship's final kinetic energy equal to this energy to find $\gamma$:
$K = (\gamma-1) ~mc^2 = 1.8\times 10^{20}~J$
$\gamma-1 = \frac{1.8\times 10^{20}~J}{mc^2}$
$\gamma = \frac{1.8\times 10^{20}~J}{mc^2}+1$
$\gamma = \frac{1.8\times 10^{20}~J}{(15,000~kg)(3.0\times 10^8~m/s)^2}+1$
$\gamma = 0.1333+1$
$\gamma = 1.1333$
Let $L_0 = 20~m$
We can find the length of the spaceship as measured by astronauts on the space station:
$L = \frac{L_0}{\gamma}$
$L = \frac{20~m}{1.1333}$
$L = 17.65~m$
The length of the spaceship as measured by astronauts on the space station is $~~17.65~m$
