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We all know that speed of light is $3\times 10^8$ $\text{ }\mathrm{ms^{-1}}$. The speed of electron inside the atom is $2.7\times 10^8$ $\text{ }\mathrm{ms^{-1}}$, which is close to speed of light.

So there is a machine that can accelerate the particle. Then when the particle move very fast, the resultant velocity of electron inside the particle should exceed the speed of light right?

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In special relativity, velocities $u$ and $v$ are "summed" according to formula $$ v_{\text{total}} = \frac{u+v}{1+\frac{uv}{c^2}}. $$ So, for example if $u=v=c$, we get $$ v_{\text{total}} = \frac{u+v}{1+\frac{uv}{c^2}}= \frac{c+c}{1+\frac{c\cdot c}{c^2}}= \frac{2c}{1+1} =c $$ As you can see, you will always get a total velocity lower or equal to $c$.

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    $\begingroup$ You seem to have dotted one of your c's $\endgroup$ – Triatticus Dec 1 '20 at 15:40

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