B Regarding the Galilean transformation of x'=x-vt

1. May 30, 2017

Ricky Pang

Hello everyone,
I am confused with the minus sign of x'=x-vt. When there are 2 references frames called K and K' which K is at rest and K' moves to right with velocity V with respect to K. Let there is another frame which is my frame of reference called O. The vector sum of the displacement should be OK'=OK+vt which equals x'=x+vt. However, this is wrong. So, I want to ask that what is the physical meaning of the minus sign of Galilean Transformation? Besides, can we apply the concept of relative motion to derive the Galilean Transformation?

2. May 30, 2017

Staff: Mentor

I am in a car moving at 100 km/hr down the road; call the frame in which I am at rest K' and the frame in which the road is at rest K. I pass a house along the road; one hour later, where is that house? It is 100 kilometers behind me, and that's what that negative sign in the $-vt$ term is saying.

More generally:
K' is moving to the right relative to frame K, so K and anything at rest in K is moving to the left with speed $v$ when considered from K'.
Consider an object that is at rest at position 0 in frame K; at time $t$ its coordinates in that frame will be $(0,t)$. However, it is moving to the left in K' so its position coordinate will become more negative with time; at time $t$ its coordinates in the primed frame will be $(-vt,t)$.

3. May 30, 2017

PeroK

The minus sign means that the velocity (if positive) represents a frame moving to the right.

4. May 30, 2017

haushofer

Just my 2 cents: another way is to differentiate the relation x'=x-vt with respect to time t. You then see that the velocity the frame x' is moving plus v equals the velocity the frame x is moving.