This abstract started with the fundamental concept called "the frame of reference".
We know that the laws of physics are not same for all observers from the simple fact that a moving observer would explain things in a different way from a relative stationary observer (unless and until they both move relatively at constant speed).
Try it for yourself by explaining the motion of objects around you i.e. for example one from the train and one from the ground.
Since physics is the most compact mathematical tool we can afford.. We don't want to create new and separate laws for moving observers and stationary observers.
And so, Galilean transformation rules are invented for the transformation of physical quantities from one reference frame to another.
Galilean transformation rules are just basic arithmetic thing. Let us consider two reference frame relatively moving at constant speed i.e. S and S' moving with relative speed "v" where else the movement considered only to be in the x-direction.
*Don't bother about the direction of motion. We can always chose the co-ordinate system such a way that x-axis lies in the direction of motion.
If a position is to be specified in the universe in "S- frame" the components are (x,y,z,) at time - t. Similarly the components measured in " S' -frame " are (x',y',z') at time - t. In Galilean time, time was considered to be the same for everyone.
If S' is thought to be moving in the positive x-direction relative to S then the components observed by two frames are related by,
x' = x -vt
y'= y
z'= z
the reverse is of course x = x' + v t ; y = y'; z = z'
Of course there should be primed coordinates on the right [because we are relating the measurements made from one frame to other.]
i.e. x = x' + v t'
but we assumed t = t' and so the problem solved.
You may ask why I am concerned about just positions?
After all, positions and their change with respect to time are just needed to explain any motion from Newton's laws of motion.
Now, Let us check how Newton's laws are behaving in these reference frames. In S - frame of reference
$$ \vec{F} = m \frac{d^2x}{dt^2} $$ Applying the same law in S' - frame, it gives $$ \vec{F'} = m \frac{d^2x'}{dt^2}$$
To compare the S' frame with S , substituting the value of x' in terms of co-ordinates observed in S-frame..$$ \vec{F'} = m \frac{d^2(x-vt)}{dt^2}$$ which gives $$ \vec{F'} = \frac {d^2x}{dt^2} - \frac{d^2(vt)}{dt^2}$$ "vt" term cancels on double integration $$ \rightarrow \vec{F'} = m \frac{d^2x'}{dt^2} = \vec{F} = \frac{d^2x'}{dt^2}$$
As we can see that, both the observers obtain the same Newtonian force and will explain any phenomena exactly in the same way. They can't even know, which one is moving from the observed motion.
Newton's laws alone cannot determine exactly either which one is moving when they move at relatively constant speed.
Thus it was realized that reference frames moving with relatively constant speed has special meaning than the usual ones and they are called "inertial reference frames".
We know that the laws of physics are not same for all observers from the simple fact that a moving observer would explain things in a different way from a relative stationary observer (unless and until they both move relatively at constant speed).
Try it for yourself by explaining the motion of objects around you i.e. for example one from the train and one from the ground.
Since physics is the most compact mathematical tool we can afford.. We don't want to create new and separate laws for moving observers and stationary observers.
And so, Galilean transformation rules are invented for the transformation of physical quantities from one reference frame to another.
Galilean transformation rules are just basic arithmetic thing. Let us consider two reference frame relatively moving at constant speed i.e. S and S' moving with relative speed "v" where else the movement considered only to be in the x-direction.
*Don't bother about the direction of motion. We can always chose the co-ordinate system such a way that x-axis lies in the direction of motion.
If a position is to be specified in the universe in "S- frame" the components are (x,y,z,) at time - t. Similarly the components measured in " S' -frame " are (x',y',z') at time - t. In Galilean time, time was considered to be the same for everyone.
If S' is thought to be moving in the positive x-direction relative to S then the components observed by two frames are related by,
x' = x -vt
y'= y
z'= z
the reverse is of course x = x' + v t ; y = y'; z = z'
Of course there should be primed coordinates on the right [because we are relating the measurements made from one frame to other.]
i.e. x = x' + v t'
but we assumed t = t' and so the problem solved.
You may ask why I am concerned about just positions?
After all, positions and their change with respect to time are just needed to explain any motion from Newton's laws of motion.
Now, Let us check how Newton's laws are behaving in these reference frames. In S - frame of reference
$$ \vec{F} = m \frac{d^2x}{dt^2} $$ Applying the same law in S' - frame, it gives $$ \vec{F'} = m \frac{d^2x'}{dt^2}$$
To compare the S' frame with S , substituting the value of x' in terms of co-ordinates observed in S-frame..$$ \vec{F'} = m \frac{d^2(x-vt)}{dt^2}$$ which gives $$ \vec{F'} = \frac {d^2x}{dt^2} - \frac{d^2(vt)}{dt^2}$$ "vt" term cancels on double integration $$ \rightarrow \vec{F'} = m \frac{d^2x'}{dt^2} = \vec{F} = \frac{d^2x'}{dt^2}$$
As we can see that, both the observers obtain the same Newtonian force and will explain any phenomena exactly in the same way. They can't even know, which one is moving from the observed motion.
Newton's laws alone cannot determine exactly either which one is moving when they move at relatively constant speed.
Thus it was realized that reference frames moving with relatively constant speed has special meaning than the usual ones and they are called "inertial reference frames".
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