Galilean Transformation: The relation of one Inertial
Frame to another Inertial Frame
If the position of a point relative to one inertial frame is given, the equation for finding the position of the same point as is determined in another inertial frame is called the Galilean transformation. The transformation equations of physical quantities from one inertial frame to another inertial frame are called the equations of the Galilean transformation.
(a) Transformation of position: Let the position vector of a point \( P \) be in inertial frame \( S \) be \( \vec{r} \). Then
where \( x,\; y,\; z \) are the coordinates of the point \( P \) and \( x',\; y',\; z' \) are the coordinates of the same point in frame \( S' \).
(b) When both inertial frames \( S \) and \( S' \) are mutually accelerated: If another inertial frame \( S' \) is moving with constant acceleration \( \vec{a} \) relative to one and the same inertial frame of reference, then the position vector of the same point relative to the two frames will be related as
or in component form \( x' = x - \dfrac{1}{2}at^{2},\quad y' = y,\quad z' = z \).
Let the position vectors of a point \( P \) in frames \( S \) and \( S' \) be \( \vec{r} \) and \( \vec{r}' \) respectively. Then
This equation (19) is called the Galilean transformation equation of position vector.
The components of the velocity \( V_{x},\; V_{y},\; V_{z} \) in reference frame \( S \), and \( V_{x}',\; V_{y}',\; V_{z}' \) respectively, then
Substituting equations (17), (19) and (20) we get
(c) Transformation of displacement: If a vector \( \vec{r} \) is given by
The displacement vector between two points is given by
Thus the vector distance between the two points is given by
From equation (20), it is evident that the displacement is same between two points from different frames of reference. This is called law of invariance of displacement of particles.
(d) Transformation of velocity: If a point \( P \) has position vector \( \vec{r} \) in \( S \), it is given more than differentiated by equation (19) with respect to time, we get
or
where \( \vec{u}' \) and \( \vec{u} \) are the velocities of particle in frames \( S' \) and \( S \).
(e) Transformation of acceleration: Differentiating equation (21) with respect to time \( t \),
Equation (22) is called Galilean acceleration transformation. It is evident from this equation that accelerations does not change when it is viewed from different reference frames. In other words, if no force is acting on the particle in one inertial frame, then the force will also be zero in another inertial frame. Newton’s first law (on which is also called law of inertia) remains valid in all reference frames which are moving with constant velocity relative to each other. According to Newton’s second law, force is mass times acceleration. Acceleration of particle does not depend on the reference frame, therefore it remains unchanged. Taking the mass \( m \) as same in the two frames, we have from Newton’s second law
Thus force acting on the particle in all inertial frames remains same. This is called law of Galilean invariance of force. (If the forces are only due to mutual interaction two systems valid in all inertial frames.)
Law of conservation of momentum: If the masses of two particles are \( m_{1} \) and \( m_{2} \) be in stationary reference frame \( S \) and if they are moving with velocities \( \vec{u}_{1} \) and \( \vec{u}_{2} \) before collision and after collision they are moving with velocities \( \vec{v}_{1} \) and \( \vec{v}_{2} \). According to the law of conservation of momentum in reference frame \( S \), we have
Let the velocities of two particles in reference frame \( S' \) before collision be \( \vec{u}_{1}' \) and \( \vec{u}_{2}' \) and after collision be \( \vec{v}_{1}' \) and \( \vec{v}_{2}' \). The system and the reference frame \( S' \) is moving with velocity \( \vec{v} \) with respect to frame \( S \). Since these two frames are inertial frames,
Total momentum in reference frame \( S' \) before collision \( = m_{1}\vec{u}_{1}' + m_{2}\vec{u}_{2}' \) (14)
Total momentum in reference frame \( S' \) after collision \( = m_{1}\vec{v}_{1}' + m_{2}\vec{v}_{2}' \)
But from Galilean transformation,
\( \vec{u}_{2}' = \vec{u}_{2} - \vec{v} \)
\( \vec{v}_{1}' = \vec{v}_{1} - \vec{v} \)
\( \vec{v}_{2}' = \vec{v}_{2} - \vec{v} \)
and
Substituting these values in equation (14), we have
or
Momentum in reference frame \( S \) before collision = Momentum in reference frame \( S \) after collision.
Thus law of conservation of momentum is valid in all inertial reference frames.
Thus Law of conservation of energy: In stationary reference frame \( S \),
Total kinetic energy before collision \( = \dfrac{1}{2}m_{1}u_{1}^{2} + \dfrac{1}{2}m_{2}u_{2}^{2} \)
Total kinetic energy after collision \( = \dfrac{1}{2}m_{1}v_{1}^{2} + \dfrac{1}{2}m_{2}v_{2}^{2} \)
According to the law of conservation of energy,
The law of conservation of energy is valid in all other inertial reference frames.
According to Galilean transformation, substituting equation (19) in equation (14), we have
Now for a vector \( \vec{a} \) and \( \vec{b} \),
and
\( \therefore\ \dfrac{1}{2}m_{1}(u_{1}'^{2} + v^{2} + 2\vec{u}_{1}'\cdot\vec{v}) + \dfrac{1}{2}m_{2}(u_{2}'^{2} + v^{2} + 2\vec{u}_{2}'\cdot\vec{v}) \)
\( = \dfrac{1}{2}m_{1}(v_{1}'^{2} + v^{2} + 2\vec{v}_{1}'\cdot\vec{v}) + \dfrac{1}{2}m_{2}(v_{2}'^{2} + v^{2} + 2\vec{v}_{2}'\cdot\vec{v}) \)
\( = \dfrac{1}{2}m_{1}v_{1}'^{2} + \dfrac{1}{2}m_{2}v_{2}'^{2} + \dfrac{1}{2}(m_{1}+m_{2})v^{2} + (m_{1}\vec{v}_{1}' + m_{2}\vec{v}_{2}')\cdot\vec{v} \)
or
(using the law of conservation of momentum).
This equation is same as equation (19). Therefore it proves that law of conservation of energy is also valid in all inertial frames of reference.