Adding velocities

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Suppose a spacecraft is moving at the velocity v vs. the system S, and that an object inside the spacecraft is moving forward with the velocity u relative to the spacecraft. What velocity, w, will the object have relative to S?

So

\displaystyle w=\frac{x}{t}

Now to an important principle we have already used. Any formula that works from S to S´ will also work from S´to S.. We only have to take the directions of the motions into consideration. That means that we can write the Lorentz transformation from S´ to S as

\displaystyle \left\{ {\begin{array}{*{20}{c}} {x=\gamma ({x}'+v{t}')} \\ {{y}'=y} \\ {{z}'=z} \\ {t=\gamma \left( {{t}'+\frac{{v{x}'}}{{{{c}^{2}}}}} \right)} \end{array}} \right.

Applying those to the formula for w, we get

\displaystyle w=\frac{x}{t}=\frac{{\gamma \left( {{x}'+v{t}'} \right)}}{{\gamma \left( {{t}'+\frac{{v{x}'}}{{{{c}^{2}}}}} \right)}}=\frac{{{x}'+v{t}'}}{{t\prime +\frac{{v{x}'}}{{{{c}^{2}}}}}}

Dividing both the numerator and denominator by t´ gives us

\displaystyle w=\frac{x}{t}=\frac{{\frac{{{x}'}}{{{t}'}}+v}}{{1+\frac{{v\frac{{{x}'}}{{{t}'}}}}{{{{c}^{2}}}}}}

But x´/t´ is u, so this gives us

\displaystyle w=\frac{{u+v}}{{1+\frac{{uv}}{{{{c}^{2}}}}}}

  1. A spacecraft moves at 0.8c vs the Earth. Two particles are fired inside the spacecraft. Both with the speed 0.7c, one forward, and one towards the back of the ship. What velocities will the two particles have relative to the Earth?
  2. What happens if you add the speed of light to the speed of light?

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