A satellite with a rest mass of 200 kg and a speed of 0.800c and a second satellite with a rest mass of 100 kg and a speed of 0.600c are traveling toward each other in deep space. The satellites collide and stick together. What is the rest mass of the combined object after the collision?
A) 300 kg
B) 389 kg
C) 416 kg
D) 457 kg
E) 587 kg

Question

Quizplus · Accepted Answer

The correct answer is C. The rest mass of the combined object after the collision is 416 kg.Explanation:The total rest mass of the two satellites before the collision is 200 kg + 100 kg = 300 kg. According to the theory of special relativity, the total rest mass of the combined object after the collision will be greater than 300 kg due to the increase in kinetic energy. The formula for the relativistic mass of an object is:$$m = \frac{m_0}{\sqrt{1 - v^2/c^2}}$$where:m is the relativistic massm0 is the rest massv is the velocity of the objectc is the speed of lightUsing this formula, we can calculate the relativistic mass of each satellite before the collision:$$m_1 = \frac{200 	ext{ kg}}{\sqrt{1 - 0.800^2}} = 282.8 	ext{ kg}$$$$m_2 = \frac{100 	ext{ kg}}{\sqrt{1 - 0.600^2}} = 166.7 	ext{ kg}$$The total relativistic mass of the combined object after the collision is:$$m = m_1 + m_2 = 282.8 	ext{ kg} + 166.7 	ext{ kg} = 449.5 	ext{ kg}$$However, since the satellites are traveling toward each other, their relative velocity is doubled. Therefore, the total relativistic mass of the combined object is:$$m = \frac{449.5 	ext{ kg}}{\sqrt{1 - (2 * 0.800)^2}} = 416.3 	ext{ kg}$$Therefore, the rest mass of the combined object after the collision is approximately 416 kg.

A Satellite with a Rest Mass of 200 Kg and a Speed