Question

How much energy is required to accelerate a spaceship with a rest mass of 123 metric tons to a speed of 0.559 c?

Every day our Earth receives 1.55×10²² J energy from the Sun. If we were able to use 1.05 percent of this energy to accelerate spaceships, then how many missions would be possible in one year?

Answer 1

Given
rest mass of spaceship is m = 123 *10^3 kgs, to accelerate with a speed of v = 0.559 *3*10^8 m/s

the energy required is equal to work done

   W = mc^2 /(sqrt(1-(v^2/c^2))) - mc^2

   = mc^2((1/sqrt(1-(v^2/c^2)))-1)
  
   = (123*10^3*(3*10^8)^2)((1/sqrt(1-(0.559^2*c^2/c^2)))-1) J

   = 2.2807378562542*10^21J

      
Earth receiving energy form sun per day is 1.55×10^22 J,

   1.5% of 1.55×10^22 J is = 2.325*10^20 J
so the the amount of energy receiving from sun in one year is = 2.325*10^20*365 J = 8.48625*10^22 J

the number of possible missions is

       n= (8.48625*10^22)/(2.2807378562542*10^21) = 37.21
  

so the number of misssions are 37