1

Let’s say an observer is travelling at 5% the speed of light. In the opposite direction a projectile approaches the observer at 99.9% light speed. According to the formulas for time dilation and length contraction, this is almost negligeable at 5% light speed. Since this is the case, shouldn’t the observer see the projectile going almost 105% the speed of light? Of course the observer will see it go slightly below light speed, but why? What physical phenomenon is causing the observer (who is hardly experiencing any time dilation or length contraction) to still see the projectile going no faster than light speed?

Luppo_Alberts
  • 19

2 Answers2

17

According to the formulas for time dilation and length contraction, this is almost negligeable at 5% light speed.

While that is true it is not relevant. What you are discussing with the projectile is not time dilation or length contraction, but rather velocity addition. The relativistic velocity addition formula is $$\frac{u+v}{1+uv/c^2}$$ We can do a second-order Taylor series expansion to get $$\frac{u+v}{1+uv/c^2}=u+v-\frac{u^2 v}{c^2} - \frac{u v^2}{c^2} + O\left(\frac{v^3}{c^3},\frac{u^3}{c^3}\right)$$ In this expansion, even if $u\approx 0$ if $v\approx c$ then the relativistic velocity addition formula is not well approximated by the Galilean $u+v$.

Dale
  • 117,350
-4

Let's say projectile speed measuring device consists of two clocks and a ruler.

After changing the speed of this device by 0.05 c, it is necessary to do a adjustment of the clocks.

This adjustment is a large one. It is large enough to correct the 5% error that occurs without the adjustment. Here 'error' means the difference of measured speed and the 'correct' speed.

stuffu
  • 2,374