Question

Assume that 437 days is a reasonable limit for how long a human can endure constant-velocity space travel. Proxima Centauri, the star closest to our Sun, is 4.24 light years away from Earth.

If you wanted to fly to Proxima Centauri within the 437-day limit in a rocket of mass 2.40×10⁶ kg , how much energy would be required to accelerate the rocket to the necessary speed in the Earth reference frame? For this rough estimate, ignore the energy required to stop, and disregard the hundreds of days required at each end of the journey for acceleration. (I tried 1.35*10^24J but it does not seem to be correct)

Compare your answer with 0.6× 1021J, the yearly worldwide energy consumption predicted for 2015.

Answer 1

Let the distance between Earth and Proxima Centauri be d which is 4.24 light years

In rocket's frame the distance is

where v is the speed of the rocket

The time taken to travel is

Hence

or

Hence

The energy of the rocket with mass m flying at this speed is

Now put the values

This gives

and the energy is

This is

times higher than the  yearly worldwide energy consumption predicted for 2015.