A rocket cruising past earth at 0.800c shoots a bullet out the back door,opposite the rocket's motion,at 0.900c relative to the rocket. What is the bullet's speed relative to the earth?
.357c
To find the bullet's speed relative to the Earth, we need to apply the relativistic velocity addition formula. This formula is given by:
v = (u + v) / (1 + (u * v) / c^2)
Where:
v = velocity of the bullet relative to Earth
u = velocity of the bullet relative to the rocket
v = velocity of the rocket relative to Earth
c = speed of light in a vacuum
Given:
u = 0.900c (bullet's velocity relative to the rocket)
v = -0.800c (rocket's velocity relative to Earth)
Plugging in the values:
v = (0.900c + (-0.800c)) / (1 + (0.900c * -0.800c) / c^2)
Now, let's solve this equation:
v = (0.1c) / (1 - 0.7200)
v = (0.1c) / (0.2800)
v = 0.3571c
Therefore, the bullet's speed relative to the Earth is approximately 0.3571 times the speed of light.
To find the bullet's speed relative to the Earth, we can use the principle of velocity addition known as the relativistic velocity addition formula.
In this scenario, we have the velocity of the rocket relative to the Earth (0.800c) and the velocity of the bullet relative to the rocket (0.900c). We want to find the bullet's velocity relative to the Earth.
The relativistic velocity addition formula is given by:
v = (u + v) / (1 + (uv / c²))
Where:
v is the velocity of an object relative to the Earth.
u is the velocity of an object relative to another object (in this case, the bullet's velocity relative to the rocket).
v is the velocity of the rocket relative to the Earth.
c is the speed of light in a vacuum.
Substituting the values into the formula, we have:
v = (0.800c + 0.900c) / (1 + (0.800c * 0.900c) / c²)
v = (1.700c) / (1 + 0.720c² / c²)
v = (1.700c) / (1 + 0.720)
v = (1.700c) / 1.720
Simplifying, we get:
v ≈ 0.988c
Therefore, the bullet's speed relative to the Earth is approximately 0.988 times the speed of light, or 0.988c.