Name: Centrifugal Force
Uploaded: 2019-07-19
Duration: 3 min 19 s
Description: Have you seen the mud flying off from a tyre? How does it happen? Have you ever think of it? Let us see one such force cause the mud to move away from the ty...

Question

Prove that the velocity of particle is observed to be different from different inertial frames, but acceleration has the same value.

Accepted Answer

Let S and S' be two inertial reference frames.

Let the position vector of the particle P with respect to S be $\vec{r}$ and position vector of P with respect to S' be $\vec{r'}$ . Let the position vector of the origin of the reference frame S' with respect to S be $\vec{r_{o}}$ . Using triangle law of addition, we can write:

${\vec{r}}_{o} + \vec{r}' = \vec{r}$ .....(i)

Differentiating both sides w.r.t. time:

$\frac{d {\vec{r}}_{o}}{d t} + \frac{d {\vec{r}}^{'}}{d t} = \frac{d \vec{r}}{d t}$

$\Rightarrow {\vec{v}}_{o} + {\vec{v}}^{'} = \vec{v}$ ........(ii)

Where $v$ is the velocity of the particle w.r.t the reference frame S and $v^{'}$ is the velocity of particle w.r.t the reference frame S'. $v_{o}$ is the velocity of the reference frame S' with respect to S. As the reference frame S' is an inertial reference frame, its acceleration with respect to S must be equal to $0$ .

Differentiating both sides of (ii) w.r.t time, we get:

$\frac{d {\vec{v}}^{'}}{d t} = \frac{d \vec{v}}{d t}$

${\vec{a}}^{'} = \vec{a}$ ......(iii)

Hence, the acceleration of the particle has the same value, when observed from the two different inertial frames. From (ii) it is clear that the particle has a different velocity when observed from different inertial reference frames.

Prove that the velocity of particle is observed to be different from different inertial frames, but acceleration has the same value.

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