# A plane flies over the head of

A plane flies over the head of Paul, an Earth observer. TreatEarth and the plane as inertial frames. The plane’s speed is v.When the plane is directly overhead, a light signal is emitted fromthe center of the plane. It is subsequently detected by twoobservers, Jack in the front of the plane, and Ben in the back ofthe plane. They both measure the distance to the center of theplane 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 reachesthem 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 atthe front and back of the plane. Why would Paul argue that Jack andBen’s detections were not simultaneous?

a)Jack and Ben are in the rest frame of the plane. So, theymeasure the distance to the center of the plane from their end as aconstant and the source at center is not moving with respect tothem. Since, the velocity of the light is c as measured by both theobservers, the time light signal takes to reach them isdistance/velocity both of which are measured equal to bothobservers. They are standing at same distance from a static sourceemitting a pulse of velocity c towards them, so they will observethe time taken by the pulse to reach them is the same and thusconclude the light reached them simultaneously.

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

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