Consider an object of mass m at rest in frame S acted on by

Consider an object of mass m at rest in frame S acted on by a ofrce F with components F_x and F_y. System S\' moves with instantaneous velocity v in the +x direction. Show that F_x\' = F_x F_y\' = F_y/gamma

Solution

Suppose that the position vector of our point object is
r = (x, y, z) in the first frame of reference. At anygiven time, t, the coordinates of the object in the two reference frames satisfy

x=x\'+ut

y=y\'

z=z\'

The instantaneuous velocy of an object in second frame v=dr/dt

so the velocity component are

Vx=Vx\'+Vx-U

Vy= Vy\'

Vz=Vz\'

similarly for acceleration a=a\'

Thefore the force Fx\'=Fx