16 | December | 2023 | Physics Town

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He can deduce the relative velocity $v_2 - v_1$ by the separations $x_2(t) - x_1(t)$ and $x_2(0) - x_1(0)$ . However, he still cannot deduce $v_2$ nor $v_1$ unless he is able to look outside the car window. Thus, he cannot deduce the car speed $v$ just by observing the positions and velocities of the objects inside the car.

For simplicity, assume object 1 is actually a point of the car itself. So $v_1$ is actually the speed of the car, $v$ . Then the calculation

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

becomes

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

where $x$ is a point of the car.

In this case, $x_{2c} = x_2(t) - x(t)$ becomes the position of object 2 relative to car; and $v_{2c} = v_2 - v$ becomes the velocity of object 2 relative to the car. The equation can be simplified to

$\displaystyle{ \begin{aligned} x_{2c}(t) &= x_{2c}(0) + v_{2c} t \\ \end{aligned}}$

If the car has acceleration, the story is totally different. In short, for the observer inside the car, the path of each particle is not a straight line anymore. In long, the previous calculation becomes

$\displaystyle{ \begin{aligned} &x_2(t) - x(t) \\ &= \Bigl( x_2(0) + v_2 t - 1/2 a t^2 \Bigr) - \Bigl(x(0) + v t - 1/2 a t^2 \Bigr) \\ &= \Bigl( x_2(0) - x(0) \Bigr) + \Bigl( v_2 - v \Bigr) t \\ \end{aligned}}$

where $a$ is the acceleration of the car. Here, we assume that $v_1, v_2,$ and $a$ are all pointing in the same direction.

Although the result is the same as before:

$\displaystyle{ \begin{aligned} x_{2c}(t) &= x_{2c}(0) + v_{2c} t, \\ \end{aligned}}$

the velocity $v_{2c}$ is no longer a constant; it would keep decreasing.

In the no acceleration case, even if the particle velocity and the car velocity are not in parallel, the observer will see a straight path. However, in the accelerated case where the acceleration and velocity directions are not in parallel, the path of the particle will no longer be a straight line.

That is what “acceleration is absolute” means. The observer can notice different phenomena, compared with the no-acceleration case, even without seeing outside the car window.

The additional meaning of “acceleration is absolute” is that we deduce the acceleration value by measuring $v_{2c}(0)$ , $v_{2c}(t)$ , and $t$ . And, unlike velocity values, this acceleration value is identical for any two observers related by a Galilean transformation. (They are called inertial observers.)