Let S be an inertial reference system. Use Galileo’s velocity addition rule.

(a) Suppose that$\overline{\mathbf{S}}$moves with constant velocity relative to S. Show that$\overline{\mathbf{S}}$is also an inertial reference system. [Hint: Use the definition in footnote 1.]

(b) Conversely, show that if$\overline{\mathbf{S}}$is an inertial system, then it moves with respect to S at constant velocity.

(a)

Using Galileo’s velocity addition rule, write the equation for the velocity of a particle with respect to S.

$u=\overline{u}+v$

Here, v is the velocity of the frame$\overline{s}$ with respect to S, u is the speed of a particle with respect to S, and $\overline{u}$is the velocity of a particle with respect to $\overline{S}$.

From the above equation, as u and v are constants, the velocity of a particle with respect to$\overline{S}$$\left(\overline{u}\right)$will also be constants. Hence,

$\overline{u}=u-v$

Hence, Newton’s law is valid in the frame$\overline{S}$.

Therefore, the frame$\overline{S}$is also an inertial frame.

(b)

When the frame is an inertial frame, the velocity of the particle in the frame $\overline{S}$will be constant.

Rearrange the equation role="math" localid="1656069585995" $\overline{u}=u-v$in terms of v.

$v=u-\overline{u}$

As the velocity of a particle with respect todata-custom-editor="chemistry" $\overline{\mathrm{S}}$is constant, the values of u and $\overline{u}$will also be constant.

Therefore, the frame$\overline{S}$ moves with uniform velocity with respect to frame S.