Question

A particle moves along the x^' axis of frame S^' with velocity 0.40c. Frame S^' moves with velocity 0.60c with respect to frame S. What is the velocity of the particle with respect to frame S?

Short answer

The velocity of the particle with respect to frame S is 0.81c.

Step-by-step solution

Describe the expression for the velocity of the particle

Suppose a particle is moving with speed u^' in x^' in an inertial frame S^'. If the frame S^' is moving with a velocity v with respect to another frame S, then the velocity of the particle with respect to frame S is given by,

$\mathbf{u}\mathbf{=}\frac{{\mathbf{u}}^{\mathbf{\text{'}}}\mathbf{v}}{\mathbf{1}\mathbf{+}{\displaystyle \frac{{\mathbf{u}}^{\mathbf{\text{'}}}\mathbf{v}}{{\mathbf{c}}^{\mathbf{2}}}}}$ …… (1)

Here, the speed of the light is c.

Determine the velocity of the particle with respect to frame S

Substitute 0.4c for u^', and 0.6c for v in equation (1).

$\begin{array}{rcl}\mathrm{u}& =& \frac{\left(0.4\mathrm{c}\right)+\left(0.6\mathrm{c}\right)}{1+{\displaystyle \frac{\left(0.4\mathrm{c}\right)\left(0.6\mathrm{c}\right)}{{\mathrm{c}}^2}}}\\ & =& \frac{1\mathrm{c}}{1.24}\\ & =& 0.806\mathrm{c}\\ & \approx & 0.81\mathrm{c}\end{array}$

Therefore, the velocity of the particle with respect to frame S is 0.81c.