Frame S’ moves relative to frame S at 0.25 c in the direction of increasing x. In frame S’, a particle is measured to have a velocity of 0.8 c in the direction of increasing x’. The velocity of the particle as measured in frame S is:

Concept:

(Relativistic) Addition of velocities or velocity addition theorem:

The Lorentz transformation equations enable us to transform velocity from one frame of reference to another, in relative motion with respect to it.

Let S and S’ be the two inertial frames in relative motion so that S; moves with uniform velocity v to the right, along the X-axis, relative to S.

Let u and u’ be the velocities of a particle measured in the inertial frames S and S’ respectively.

\(u' = \frac{{u - v}}{{1 - \frac{{uv}}{{{c^2}}}}}\)

So, if a particle has a velocity u’ in a frame of reference S’ which is moving with velocity v relative to another frame of reference S, the velocity of a particle in the frame S will be:

\(u = \frac{{u' + v}}{{1 + \frac{{u'v}}{{{c^2}}}}}\)

Calculation:

Given:

Frame S’ moves relative to frame S at 0.60 c in the direction of increasing x. In frame S’, a particle is measured to have a velocity of 0.4 c in the direction of increasing x’. The velocity of the particle as measured in frame S is:

v = 0.25 c, u’ = 0.8 c

\(u = \frac{{u' + v}}{{1 + \frac{{u'v}}{{{c^2}}}}} \)

\(= \frac{{\left( {0.25 + 0.8} \right)c}}{{1 + \frac{{0.25 \times 0.8{c^2}}}{{{c^2}}}}} = \frac{{\left( {0.25 + 0.8} \right)}}{{1 + 0.25 \times 0.8}}c = 0.875c\)