Solved Problem on Relativity
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A spaceship (E) travels in space with a speed equal to 90% of the speed of light relative to the Earth, and another spaceship (K) travels towards it with a speed equal to 80% of the speed of light also relative to the Earth. Determine what is the speed of ship K relative to ship E?
Problem data:
- Velocity of ship E: vE = 0.90 c;
- Velocity of ship K: vK = 0.80 c.
We choose a reference frame R fixed on Earth, letting v as the velocity of ship K relative to Earth, vK = v = −0.80 c. We choose another fixed reference frame R' on ship E, the speed of ship E will be the same speed as the reference frame R' relative to the reference frame R fixed on Earth, and this speed will be indicated by vE = μ = 0.90 c (Figure 1).
What we want to know is the velocity of ship K relative to reference frame R', denoted by v'.
Solution
As the speeds of the ships are close to the speed of light by Einstein's Relativity, the speed of K relative to E will be
\[
\begin{gather}
\bbox[#99CCFF,10px]
{v'=\gamma (v-\mu )}
\end{gather}
\]
where
\( \bbox[#99CCFF,10px] {\gamma =\dfrac{1}{1-\dfrac{\mu v}{c^{2}}}} \)
substituting problem data
\[
\begin{gather}
v'=\frac{1}{1-\dfrac{\mu v}{c^{2}}}(v-\mu)\\[5pt]
v'=\frac{1}{1-\dfrac{0.90 c \times(0.80 c)}{c^{2}}}(-0.80 c-0.90 c)\\[5pt]
v'=\frac{1}{1-\dfrac{(-0.72 \cancel{c^{2}})}{\cancel{c^{2}}}}(-1.70 c)\\[5pt]
v'=\frac{1}{1+0.72}(-1.70 c)\\[5pt]
v'=\frac{-1.70 c}{1.72}
\end{gather}
\]
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{v'=-0.98c}
\end{gather}
\]
Note 1: The negative sign in the answer indicates that the direction of the velocity of
ship K is in the opposite direction to the reference frame R' (Figure 1).
Note 2: Contrary to what happens in Galileo's Relativity, where the relative velocity is the sum of the velocities
Note 2: Contrary to what happens in Galileo's Relativity, where the relative velocity is the sum of the velocities
\[
\begin{gather}
v'=v-\mu \\[5pt]
v'=-0,80c-0,90c\\[5pt]
v'=1,70c
\end{gather}
\]
in Einstein's Relativity, this does not happen because we cannot have speeds greater than the
speed of light.
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .