## Galilean Relativity

At the basis of Newtonian mechanics lies an idea we call galilean relativity, which is the idea that different observers will come up with the same laws of motion as long as their reference frames are not accelerating relative to each other. The reason that acceleration is ‘special’, is because Newton defined force in terms of acceleration: If force equals mass times acceleration, then an accelerating observer will see other forces than a non-accelerating observer.

When two people see something happening, and one of them is moving while the other isn’t, they will see different things. Imagine people playing a game of tennis inside a moving ship, and compare them with someone on the shore looking at the game.

Differently moving observers see different things, yet these observers agree that the laws of physics are the same, and that momentum is conserved – How can that be?

Of course, for the person on the shore the ball seems to move faster when moving in the same direction as the ship, but it also gets an extra push whenever it starts moving in that direction (because it already has some speed in that direction). From the point of view of the onlooker on the shore, all motion in the tennis game is shifted (transformed) in the same way.

The tennis players and the onlooker disagree on the motion that takes place, but they nevertheless agree on the laws of motion. In their own reference frame they see different velocities, but the acceleration seen is the same in the different reference frames. To see how that works, we consider an example.

Suppose the tennis ball is moving at a speed of 60 m/s in the same direction as the ship, which itself moves with 10 m/s, so the onlooker on the shore sees the ball whizzing by at 70 m/s. Now suppose the ball is hit with a tennis racket so that its speed increases from 60 to 80 m/s. That would mean, in the reference frame of the tennis players, an acceleration of 20 m/s. The onlooker on the shore, however, would see the ball fly at 70 m/s before it gets hit by the racket. But that racket, in the reference frame of the onlooker on the shore – and here’s the great trick – also has a greater speed than it has in the reference frame of the tennis players, so it will hit the ball harder. How much harder? The speed of the racket gets added to the speed of the ship, which was 10 m/s. So the ball’s speed increases from 70 to 90 m/s. We see that the acceleration is 20 m/s, regardless of whether we look at the situation in the reference frame of an observer that is standing still or an observer that is moving.

Changes of reference frame in which velocities differ but acceleration does not, are called galilean transformations. The idea that the same laws of physics hold in reference frames that are related through galilean transformations is called galilean relativity.

A familiar, everyday, example is playing with a yoyo: it doesn’t matter whether you do this at home or in a moving train; the laws we use to describe the motion of the yoyo  are the same, which means that the throwing and pulling you need to do to get the yoyo to move the way you want is also the same.

Stop and think!

• Why does galilean relativity not hold for accelerating reference frames?

Hint: think of Newton’s second law.

LEARNING OBJECTIVE:

This text should give the reader a basic idea of what the Newtonian view of physics/mechanics is, and particularly the role that acceleration and galilean relativity play in it.

## About fbenedictus

Philosopher of physics at Amsterdam University College and Utrecht University, managing editor for Foundations of Physics and international paraclimbing athlete
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