## The speed of a bullet

Imagine a marksman on earth armed with a super-rifle. He fires a shot at an astronaut who is travelling away from him in a spaceship. Luckily the shot misses its target. The astronaut, a cool customer, measures the speed of the bullet as it passes the spaceship.

The earthman and the astronaut disagree about the speed of the bullet. As far as the astronaut is concerned, the bullet is trying to overtake him and it will appear to him to move more slowly.

In terms of reference frames:

(See Figure 15.7 on next page) S: Earth frame of reference S': Astronaut's frame of reference v = velocity of S' relative to S (the velocity of the spacecraft as measured by the marksman)

Let v' = velocity of the bullet as seen by the astronaut. Let us compare the speed of the bullet as measured by the astronaut in his 'laboratory', with the speed of the same bullet as measured on earth. In the earth frame of reference S we see the astronaut moving with a velocity v in the x direction. The bullet, which is moving with a velocity V, also in the x direction, will overtake the astronaut at a relative speed of V - v. As far z

speeding bullet

Figure 15.7 The astronaut and the earthman measure the speed of the same bullet.

x as the astronaut is concerned, the speed of the bullet is V - v in his frame of reference S'.

Speed of bullet in the earth laboratory = V

Speed of bullet in the astronaut laboratory = v' = V - v

We can obtain the same result more formally by applying a Galilean transformation.

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