Chapter 15, Problem 15.1P

To determine

The proof that Newton’s laws are invariant under Galilean transformation.

Answer to Problem 15.1P

Newton’s laws are invariant under Galilean transformation.

Explanation of Solution

Write the three expressions for Galilean transformation

    $v^{\prime }=v-V$        (I)

Here, $v$ is the velocity of the object in the frame $S$, $v^{\prime }$ is the velocity in the $S^{\prime }$ frame and $V$ relative velocity between the two frames.

Differentiate equation (I)

    $\begin{array}{c}\frac{d\left(v^{\prime }\right)}{dt}=\frac{d\left(v-V\right)}{dt}\\ a^{\prime }=a\end{array}$        (II)

Write the expression for force in frame $S$

    $F=ma$        (III)

Here, $F$ is the force in frame $S$.

The net force acting on the object in frame $S$ is zero, if the acceleration is zero and the body continues to remain in it state of motion or of rest as stated by Newton’s first law.

Substitute $0$ for $a$ in equation (II)

    $a^{\prime }=0$

Write the expression for force in frame $S^{\prime }$

    $F^{\prime }=m{a}^{\prime }$        (IV)

Here, $F^{\prime }$ is the force in frame $S^{\prime }$.

Write the expression for reaction force in $S$ frame

    $F_{\text{reaction}}=-F$        (V)

Rewrite the above equation using (III)

    $\begin{array}{c}{F}_{\text{reaction}}=-ma\\ =m\left(-a\right)\end{array}$        (VI)

Thus, the acceleration of the reaction force in $S$ frame is $-a$.

Conclusion:

Substitute $0$ for $a$ in equation (IV)

    $F^{\prime }=0$

The force and acceleration in frame $S^{\prime }$ are also zero and hence the object in frame $S^{\prime }$ also remains in its state of motion. Thus, Newton’s first law is invariant under Galilean transformation.

Write the expression for reaction force in $S^{\prime }$ frame using (II) and (IV)

    $\begin{array}{c}{F}_{\text{reaction}}=m\left(-a^{\prime }\right)\\ =-m{a}^{\prime }\\ =-F^{\prime }\end{array}$        (VII)

Thus, Newton’s third law is also invariant under Galilean transformation.