# Mechanics part 4: Galilean Relativity

Welcome back fellow carbon-based life forms! Last time, we talked about momentum and its conservation. As we saw, it allows us to analyze the interactions between particles just by knowing their initial momentum! Basically, if there are no external forces on a system, its momentum is always conserved. Always.

Today I want to talk about something different yet related. Reference frames. You may have heard of them before, in relativity and stuff. But what I’m gonna talk about today isn’t Einsteinian relativity, as they call it, but Galilean relativity. This form of relativity holds for object moving at speeds much, much slower than the speed of light. So yeah, no special or general relativity today.

Let’s start with the question in your head. What exactly is a reference frame? In simple words, a reference frame is a coordinate system attached to a body. In your reference frame, you measure the position and velocities of particles relative to you.

As you can see, the tree is 40 m away relative to A but 16 m away relative to B. So now let’s generalize it to multiple dimensions. We will have two frames S and S’, attached to A and B respectively. This means that we’ll have A positioned at O at all times, and B positioned at O’ at all times. We’ll call S the ‘ground frame’, and for our purposes is stationary. Let R be the position of B, and r be the position of point P relative to A, and r’ be the position of that same point relative to B.

As you can see,

Now we assume that B is moving with some non-zero velocity. Then the position of the P changes with time relative to B. Thus

Now let’s assume there’s an object C at point P, and it also moves relative to the ground, then

So as you can see above, the velocity of C relative to B is simply its velocity relative to A minus the velocity of B relative to A. Pretty neat huh? Now let’s differentiate again, this time assuming B moves with a constant velocity relative to A. We see that

So notice what we got. If B moves with a constant velocity, then the acceleration of C with respect to B is the same as with respect to A. Taking into consideration the fact that multiplying this result by the mass of particle C will give Newton’s 2nd law, we see that the force on particle C is the same relative to both A and B (because mass doesn’t — welp, that’s a story for another time) Newton’s laws remain the same even if you switch from one inertial frame to another

This sort of reference frame, in which Newton’s laws remain the same relative to a stationary observer, is called an inertial frame. Basically the frame is moving at a constant velocity. Also, now’s a good time to point out that there is no such thing as absolute rest. This can be seen pretty easily. If you have a car moving at 50 km/hr relative to say, a tree, then the tree to moves at 50 km/hr in the opposite direction as the car, relative to the car. Absolute rest doesn’t exist.