A space tow barge pushes on a spaceship with a rest mass of 15,000 kg and accelerates it from a speed of 0.600c to a speed of 0.700c. How much work does the tow barge have to do to accomplish this? A) 1.35 × 10²⁰ J B) 2.03 × 10²⁰ J C) 2.70 × 10²⁰ J D) 5.42 × 10²⁰ J E) 8.78 × 10¹⁹ J

Question

Quizplus · Accepted Answer

We will use the relativistic kinetic energy equation:
$K = (\gamma-1)mc^2$
where $\gamma$ is the Lorentz factor given by $\gamma = \dfrac{1}{\sqrt{1-(v/c)^2}}$, $m$ is the rest mass of the spaceship, and $v$ is the final velocity of the spaceship.

At the initial velocity, $\gamma_1 = \dfrac{1}{\sqrt{1-(0.6c/c)^2}} = 1.25$.
At the final velocity, $\gamma_2 = \dfrac{1}{\sqrt{1-(0.7c/c)^2}} = 1.40$.

The work done by the tow barge is equal to the change in kinetic energy of the spaceship:
$W = K_2 - K_1 = (\gamma_2-1)mc^2 - (\gamma_1-1)mc^2$

Substituting the given values, we get:
$W = (1.40-1.25) 	imes 15000 	imes (3	imes10^8)^2 = 2.03 	imes 10^{20}$ J.

Therefore, the answer is choice (B), 2.03 × 10²⁰ J.

A Space Tow Barge Pushes on a Spaceship with a Rest