The Lorentz transformation makes it possible to know what will happen if we observe a phenomenon moving relative to the object, the carrier of the phenomenon, at a speed , or if the object moves relative to us at a speed , provided that we know how the phenomenon occurs when the object is stationary. Therefore, the following statement of the question will often be used in the future. In a stationary system, the phenomenon is described. How will this phenomenon occur if the object that carries the phenomenon is moving at speed ? The answer assumes a description of this phenomenon in the system a reference point that moves at a speed of relative to the system in which the phenomenon is described. This is equivalent to describing this phenomenon in the case of the movement of an object, the carrier of the phenomenon, with a velocity relative to a stationary observer. The second option is interesting because it can be extended to several isolated objects moving at different speeds. Use this to solve the following problem. The observation station recorded light signals from two rockets moving in a straight line towards the station. The frequencies of signals registered by the station are and . The signal frequency of stationary rockets is equal to . How fast do rockets approach each other?
Solution
Velocity of each rocket from the station
The station receives frequencies and from sources that are approaching. The Doppler effect formula for direct approachis:
Solving for
In terms of velocity:
Velocity difference in the station's frame
The rockets move along the same line toward the station. The speed with which they approach each other, measured in the station's frame, is simply the difference of their velocities (if one is faster than the other) or the sum (if they come from opposite directions). We assume they move in the same direction (one behind the other):
Discussion
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