24 | September | 2023 | Physics Town

. . .

The meaning of “velocity is relative” is:

For example, within a car, you cannot know its velocity relative to the ground without seeing outside. In other words,

You cannot know the velocity of A (car) relative to the B (ground) without seeing B (outside).

It is because any two individual objects within the car, if both initially at rest relative to the car, have a constant separation. Also, any individual objects inside the car and any point of the car itself has a constant separation. And here is the proof:

The separation between any 2 objects within the car is

$\displaystyle{ \begin{aligned} &x_2(t) - x_1(t) \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} &= \Bigl( x_2(0) + v_2 t \Bigr) - \Bigl(x_1(0) + v_1 t \Bigr) \\ \end{aligned}}$ ,

where $\displaystyle{v_1}$ and $\displaystyle{v_2}$ are velocities with respect to the ground of object 1 and object 2 respectively. If the two velocities have the same value,

$\displaystyle{ \begin{aligned} &x_2(t) - x_1(t) \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} &= x_2(0) - x_1(0) \\ \end{aligned}}$

As long as object 1 and object 2 have the same velocity-relative-to-the-ground as that of the car, $\displaystyle{v}$ , i.e.

$\displaystyle{v_1=v_2=v}$ ,

no matter what value $\displaystyle{v}$ has, the distance between object 1 and object 2 is always constant. In other words, you cannot deduce the value of the $\displaystyle{v}$ by observing the separation changes between any two objects/points within the car.

Even in another case where $\displaystyle{v_1 \ne v_2}$ ,

…

— Me@2023-08-07 05:56:31 AM

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