A rocket cruising past Earth at 0.79c shoots two bullets at 0.92c relative to the rocket with one of the bullets fired along the direction of the rocket's motion and the other fired opposite the rocket's motion.

(a) What velocity does an Earth-based experimenter measure for the bullet fired in the direction of motion of the rocket? Give your answer as a fraction of c, where positive values are used if the velocity is in the same direction of the ship.

(a) What speed does an Earth-based experimenter measure for the bullet fired in the direction of motion of the rocket? Give your answer as a fraction of c, where positive values are used if the velocity is in the same direction of the ship.

There are relativistic formulas for adding relative velocities. You need to learn them.

(a) In Earth-fixed coordinates, the bullet fired forward has a velocity
(0.79c + 0.92c)/[1 + (0.79c*0.92c/c^2]
= 1.710c/1.727 c = 0.990c

(b) In Earth-fixed coordinates, the bullet fired backwards has velocity
(0.79c - 0.92c)/[1 + (0.79c*0.92c/c^2]
= -0.13/1.727 c = -0.075c

The minus sign means it is going backwards relative to the space ship.

To determine the velocity of the bullet fired in the direction of motion of the rocket as measured by an Earth-based observer, we need to apply the principle of relativistic addition of velocities.

The formula for relativistic addition of velocities is:
v' = (v + u) / (1 + (vu / c^2))

Where:
- v' is the velocity observed by the Earth-based observer
- v is the velocity of the rocket relative to Earth (0.79c)
- u is the velocity of the bullet relative to the rocket (0.92c)
- c is the speed of light in a vacuum (approximately 3 x 10^8 m/s)

Now, let's plug in the given values into the formula and calculate the velocity of the bullet fired in the direction of motion of the rocket, as observed from Earth:

v' = (0.79c + 0.92c) / (1 + ((0.79c)(0.92c) / (c^2)))
v' = (1.71c) / (1 + (0.72748))
v' = (1.71c) / (1.72748)
v' ≈ 0.99c

Therefore, the Earth-based experimenter would measure the velocity of the bullet fired in the direction of motion of the rocket to be approximately 0.99 times the speed of light (c), in the same direction as the rocket.