# An electron moves to the right with a speed of 0.93 c relative to the laboratory frame. A proton...

## Question:

An electron moves to the right with a speed of {eq}0.93\ c {/eq} relative to the laboratory frame. A proton moves to the left with a speed of {eq}0.63\ c {/eq} relative to the electron. Find the speed of the proton relative to the laboratory frame.

## Relative Velocity in Fast Moving Systems

According to the theory of relativity, the relative velocity of a fast moving particle with respect to another moving object is given by:

{eq}\displaystyle u'=\frac{u-v}{1-\frac{uv}{c^2}} {/eq}

Where,

• {eq}u' {/eq} is the velocity of the particle with respect to the moving frame,
• {eq}u {/eq} is the velocity of the particle with respect to rest frame,
• {eq}v {/eq} is the velocity of the moving frame with respect to the rest frame and
• {eq}c=3.00\times10^8\ m/s {/eq} is the speed of light.

Given:

{eq}u=0.93c\\ u'=0.63c {/eq}

The velocity of proton with respect to laboratory frame is,

{eq}\displaystyle v=\frac{u'-u}{\frac{u'u}{c^2}-1}=\frac{0.63c-0.93c}{\frac{0.63c\times0.93c}{c^2}-1}=0.72c {/eq} 