A rocket cruising past earth at 0.800c shoots a bullet out the back door, opposite the rocket's motion, at 0.950c relative to the rocket. What is the bullet's speed relative to the earth?
To determine the bullet's speed relative to the Earth, we can use the relativistic velocity addition formula. This formula takes into account the velocity of the bullet relative to the rocket and the velocity of the rocket relative to the Earth.
The relativistic velocity addition formula is as follows:
v' = (v + u) / (1 + (v * u) / c^2),
where:
v' = final velocity relative to the Earth,
v = velocity of the rocket relative to the Earth,
u = velocity of the bullet relative to the rocket,
c = speed of light.
Given:
v = 0.800c (velocity of the rocket relative to the Earth),
u = 0.950c (velocity of the bullet relative to the rocket).
Using these values in the formula, we get:
v' = (0.800c + 0.950c) / (1 + (0.800c * 0.950c) / c^2)
= (1.750c) / (1 + 0.760)
Simplifying further:
v' = (1.750c) / 1.760
≈ 0.9943c
Therefore, the bullet's speed relative to the Earth is approximately 0.9943 times the speed of light (c).
To determine the bullet's speed relative to the Earth, we can use the principle of relativity, which states that the laws of physics are the same in all inertial reference frames.
First, let's break down the given information:
- The rocket's speed relative to the Earth is 0.800c, where 'c' denotes the speed of light.
- The bullet's speed relative to the rocket is 0.950c.
To find the bullet's speed relative to the Earth, we need to use the velocity addition formula in special relativity. The formula is:
v = (u + v) / (1 + (u * v) / c^2)
Where:
- v is the relative velocity between the Earth and the bullet
- u is the relative velocity between the rocket and the Earth
- v is the relative velocity between the bullet and the rocket.
- c is the speed of light.
Substituting the given values into the formula, we have:
v = ((0.950c) + (0.800c)) / (1 + ((0.950c) * (0.800c)) / c^2)
v = (1.750c) / (1 + (0.760c^2/c^2))
v = (1.750c) / (1 + 0.760)
v = (1.750c) / 1.760
v ≈ 0.9943c
Therefore, the bullet's speed relative to the Earth is approximately 0.9943 times the speed of light (c).