Galilean transformations These are equations that transform physical quantities measured in one reference frame to another frame. The two frames are moving with a constant relative velocity with respect to one another ( á á c). Let one frame O be taken as rest and the other O¢ moving with constant velocity V along the x - x¢ axis with respect to O. The axis y is parallel to y¢ and the axis z parallel to z¢ (see fig. g2). The two frames have clocks attached which are synchronized when O and O¢ coincide. Let at an instant of time t an observer in O see a particle of mass m at the point (x, y, z) being acted upon by a force F. The position coordinates of the same particle in O¢ will be (x¢ , y¢ , z¢ ). The relationships between the coordinates in the two frames are given by,

x¢ = x - Vt

y¢ = y

z¢ = z

t¢ = t (by assumption) (Galilean coordinate transformation)(1)

Taking time derivative of eq. above we obtain Galilean velocity transformations.

u_x¢ = u_x - V

u_y¢ = u_y

u_z¢ = u_z Galilean velocity transformation (2)

Differentiating eq. (2) once again with respect to time we obtain the acceleration transformations.

a_x¢ = a_x

a_x¢ = a_x

a_z¢ = a_z (3)

By assumption the mass of the particles is the same in the two reference frames. Then from eq.(3) the force as computed in one reference frame is equal to the force computed in the second reference frame that is moving with constant velocity with respect to the first. In other words the Newton’s laws are invariant under Galilean transformations. This is known as Galilean invariance.