Question

A plane flies over the head of Paul, an Earth observer. Treat Earth and the plane as inertial frames. The plane’s speed is v. When the plane is directly overhead, a light signal is emitted from the center of the plane. It is subsequently detected by two observers, Jack in the front of the plane, and Ben in the back of the plane. They both measure the distance to the center of the plane to be d.

a. Assume the speed of light is c as measured by Jack and Ben. Explain why Jack and Ben both agree that the light signal reaches them simultaneously.

b. Assume the speed of light is c as measured by Paul on Earth. Explain what Paul would observe about the light signal arriving at the front and back of the plane. Why would Paul argue that Jack and Ben’s detections were not simultaneous?

Answer 1

a)Jack and Ben are in the rest frame of the plane. So, they measure the distance to the center of the plane from their end as a constant and the source at center is not moving with respect to them. Since, the velocity of the light is c as measured by both the observers, the time light signal takes to reach them is distance/velocity both of which are measured equal to both observers. They are standing at same distance from a static source emitting a pulse of velocity c towards them, so they will observe the time taken by the pulse to reach them is the same and thus conclude the light reached them simultaneously.

b)Paul observes 2 moving observers and I moving source between them exactly at the middle of the plane. Let all of them move in positive x direction. So, if the distance between source and observers are L, the time required for the light to cover this distance is L/c. During this interval both the observers have moved to a new position by dx = v* L/c. Both the observers moved a distance of dx in the positive x direction. Thus, the effective distance between the source and Jack is now L/c + dx, where as for Ben it is L/c -dx . Since, the light had to move in -ve x direction to reach Ben. So, Paul observes that light reach Ben before Jack, because of the relative motion of the plane with respect to Paul.