If two rocket ships travel at 0.63 c and 0.91 c relative to the ground, how fast do they move relative to each other?

To determine the speed of one rocket ship relative to the other when they are moving at significant fractions of the speed of light, we must apply the principles of relativistic velocity addition. The formula for adding two velocities (u) and (v) that are both less than the speed of light is given by:

[

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

]

In this case, let's set ( u = 0.63c ) and ( v = 0.91c ). Plugging these values into the formula:

[

w = \frac{0.63c + 0.91c}{1 + \frac{(0.63c)(0.91c)}{c^2}}

]

[

= \frac{(0.63 + 0.91)c}{1 + 0.63 \times 0.91}

]

[

= \frac{1.54c}{1 + 0.5733}

]

[

= \frac{1.54c}{1.5733}

]

[

\approx 0.98c

]

This calculation shows