A rocket cruising past earth at 0.700 c shoots a bullet out the back door, opposite the rockets motion, at 0.900 c relative to the rocket. What is the bullets speed relative to the earth?

To determine the bullet's speed relative to Earth, we need to apply the principles of relativistic velocity addition. When dealing with speeds that are a significant fraction of the speed of light (denoted as "c"), classical mechanics no longer applies, and we must use the relativistic formulas to find the correct speed. Let's break this down step by step.

Understanding the Scenario

In this situation, we have a rocket moving at a speed of 0.700 c relative to Earth. The bullet is fired from the rocket at a speed of 0.900 c relative to the rocket, but in the opposite direction of the rocket's motion. To find the bullet's speed relative to Earth, we need to use the relativistic velocity addition formula.

The Relativistic Velocity Addition Formula

The formula for adding velocities in relativity is:

u' = (u + v) / (1 + (uv/c²))

Where:

• u' is the velocity of the bullet relative to Earth.
• u is the velocity of the rocket relative to Earth (0.700 c).
• v is the velocity of the bullet relative to the rocket (0.900 c, but since it's in the opposite direction, we will consider it as -0.900 c).

Plugging in the Values

Now, substituting the values into the formula:

u' = (0.700 c - 0.900 c) / (1 - (0.700 * 0.900))

Calculating the Numerator

First, let's calculate the numerator:

0.700 c - 0.900 c = -0.200 c

Calculating the Denominator

Next, we calculate the denominator:

1 - (0.700 * 0.900) = 1 - 0.630 = 0.370

Final Calculation

Now we can substitute these values back into the formula:

u' = (-0.200 c) / (0.370)

Calculating this gives:

u' ≈ -0.540 c

Interpreting the Result

The negative sign indicates that the bullet is moving in the opposite direction to the positive direction we defined (the direction of the rocket's motion). Therefore, the bullet's speed relative to Earth is approximately 0.540 c in the opposite direction of the rocket's travel.

Summary

In summary, when the rocket traveling at 0.700 c fires a bullet backward at 0.900 c relative to itself, the bullet's speed relative to Earth is approximately 0.540 c in the opposite direction of the rocket's motion. This example illustrates the fascinating and sometimes counterintuitive nature of relativistic physics, where speeds do not simply add together as they do in classical mechanics.