Question

according to an observer on earth,a spaceship is going east with a speed of 0.6c and is going to collide head on in 5s with a comet going west at 0.80c.How fast does the spacecraft see the comet to be approaching.According to their clocks,how much time do they have to get out of the way

Answer 1

Solution:relativistic speed addition:

$$S=\dfrac{v+u}{1+\frac{vu}{c^2}}$$

$$S=\dfrac{\frac{0.6}{c}+\frac{0.8}{c}}{1+0.6*0.8}=\dfrac{\frac{1.4}{c}}{1.48}=0.946c$$

time dilation:

$$t'=\dfrac{1}{γγ}=\dfrac{1}{\sqrt{\smash[b]{(1-(\frac{V^2}{c^2}))}}}$$

$$γ=\dfrac{1}{\sqrt{\smash[b]{(1-(0.946^2))}}}=3.08$$

$$t'=\dfrac{5}{3.08}=1.62s$$