An observer on Earth sees an alien vessel approach at a speed of 0.44c. The Enterprise comes to the rescue, overtaking the aliens while moving directly toward Earth at a speed of 0.89c relative to Earth. What is the speed of one vessel as seen by the other?

Question

An observer on Earth sees an alien vessel approach at a speed of 0.44c. The Enterprise comes to the rescue, overtaking the aliens while moving directly toward Earth at a speed of 0.89c relative to Earth. What is the speed of one vessel as seen by the other?

Answer 1

To find the relative velocity between two objects moving at relativistic speeds, we need to use the relativistic velocity addition formula. The formula for adding velocities in special relativity is:

v' = (v1 + v2) / (1 + (v1*v2)/c^2)

Where:
v' is the relative velocity between the two objects,
v1 is the velocity of one object relative to an observer,
v2 is the velocity of the other object relative to the same observer, and
c is the speed of light in a vacuum (approximately 3.00 x 10^8 meters per second).

In this scenario, we have two objects: the alien vessel and the Enterprise.

Observer's perspective:
The alien vessel is approaching Earth at a speed of 0.44c.

v1 = 0.44c

The Enterprise is moving toward Earth at a speed of 0.89c relative to Earth.

v2 = 0.89c

Now we can substitute these values into the formula to find the relative velocity between the two objects:

v' = (0.44c + 0.89c) / (1 + (0.44c * 0.89c)/(c^2))

Simplifying the equation:

v' = (1.33c) / (1 + (0.44 * 0.89))

v' = 1.33c / (1 + 0.3916)

v' = 1.33c / 1.3916

v' ≈ 0.9568c

Therefore, the speed of one vessel as seen by the other is approximately 0.9568 times the speed of light.

Answer 2

To determine the speed of one vessel as seen by the other, we can use the relativistic velocity addition formula:

v' = (v + u) / (1 + (v*u) / c^2)

Where:
v' = speed of one vessel as seen by the other
v = velocity of the observer on Earth seeing the alien vessel approach (0.44c)
u = velocity of the Enterprise relative to Earth (0.89c)
c = speed of light (299,792,458 meters per second)

Let's substitute the values into the formula and calculate:

v' = (0.44c + 0.89c) / (1 + (0.44c * 0.89c) / c^2)

First, let's simplify the expression in the denominator:

(0.44c * 0.89c) / c^2 = 0.3916

Now, substitute the simplified value and calculate:

v' = (0.44c + 0.89c) / (1 + 0.3916)

v' = (1.33c) / (1.3916)

v' = 0.956c

Therefore, the speed of one vessel as seen by the other is approximately 0.956 times the speed of light.