How to properly use Galilean transformation in order to move between inertial frames?

Question

I'm having trouble understanding how to use Galilean transform when moving between coordinates. e.g consider the following problem: a mass attached to a spring inside a moving cart - how can I represent the coordinates of the cart and the mass from the different inertial frames - the laboratory frame and the cart frame.

If I try to find the coordinates of the mass from inside the cart, I thought they should look something like (x,0) in contrast to when I look from the laboratory inertial frame:

It'll be (x+v_0 t,0) and if I want to talk about the cart's location then I can denote it's coordinates as (x', y')

how can I generalize this, I get confused sometimes between the cases where I need to add v0*t or subtract it?

Answer 1

The essential thing you need here is the relation between the displacement vectors between $2$ different reference frames.

Let the point $P$ has position vector $\vec r|_{O}$ in the reference frame $O$ and $\vec r|_{O'}$ in frame $O'$. Let the position vector of the origin of $O$ with respect to the origin of $O'$ be $\vec {O'O}$. Then, we have - $$\vec r|_{O'} = \vec r|_{O} + \vec {O'O}$$ that is, position wrt origin of $O =$ position wrt origin of $O'+$ position of origin of $O'$ wrt origin of $O'$.

So, in your case, as reference frame $O$ is moving at velocity $v$ wrt $O'$, we have $\vec {O'O} = v_0t$. Thus, $x' = x + v_0t$. The others follow similarly.