Question

Anna and bob are playing catch on a moving train. according to them, in the frame of the train, they stand 10,000 meters apart. anna throws the ball to bob at a speed 0.1c, according to her, in the positive x direction. carl watches this game of catch from the train station platform

if the train is moving at 0.5c in the positive x direction relative to the platform, how far apart are anna and bob according to carl?

Answer 1

To determine the distance between Anna and Bob as observed by Carl on the train platform, we apply the concept of Lorentz contraction due to the relativistic speeds involved. According to Carl, the distance between them, originally 10,000 meters in their frame, is observed to be approximately 8,660 meters.

Step-by-step explanation:

Relativity and Distance Perception

The scenario described is a classic example of the relativity of simultaneous events and the perception of distance in different reference frames.

To determine how far apart Anna and Bob are according to Carl, who is stationary on the train platform, we must account for Lorentz contraction, a relativistic effect that occurs at speeds close to the speed of light (c).

In Anna and Bob's frame of reference (inside the moving train), they are 10,000 meters apart.

However, due to Lorentz contraction, Carl will observe this distance to be shorter.

The Lorentz contraction formula is L = L0*sqrt(1-(v^2/c^2)), where L is the length observed by Carl, L0 is the proper length observed by Anna and Bob, v is the velocity of the train relative to Carl, and c is the speed of light.

Substituting the provided values into the Lorentz contraction formula, we get:
L = 10,000 meters * sqrt(1-(0.5c)^2/c^2)

= 10,000 meters * sqrt(1-0.25)

= 10,000 meters * sqrt(0.75)

≈ 8,660 meters.

Therefore, according to Carl, Anna and Bob are approximately 8,660 meters apart.