## Addition of velocities a classical example

A fighter plane travelling at a speed v fires bullets forwards with a muzzle velocity V. What is the resultant velocity of the bullets relative to the ground?

Classically, there is no problem. The velocity relative to the ground is the vector sum of the two velocities. Since they are in the same direction, we simply add the two velocities, giving the resultant VR = v + V.

Again classically, we may use the slightly more sophisticated classical method of applying the Galilean transformation to transform from the frame of reference of the pilot to that of the earth observer, to compare how the speed of the bullet appears to each.

V = velocity of bullet in frame S'. What is the velocity of the same bullet in frame S?

Taking all velocities to be in the x direction,

earth frame

¿V figh fighter pilot frame x = g(x - vt ) -=--v dt dt

 V dx ' dt Vr dx = dt

Classically there is no limit to the sum of the velocities. For example, it may happen that the resultant velocity of the bullet is greater than the speed of sound.

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