A spaceship at rest in a certain reference frame S is given a speed increment of 0.50c. Relative to its new rest frame, it is then given a further 0.50c increment. This process is continued until its speed with respect to its original frame S exceeds 0.999c. How many increments does this process require? (Answer: 7)

To find the number of increments required, we need to calculate how many times the spaceship's speed increases by 0.50c until it exceeds 0.999c.

Let's start by calculating the spaceship's speed after each increment. We'll use the Lorentz transformation to calculate the relative velocity.

The Lorentz transformation for velocity addition is given by:

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

Where:
v' = the resulting velocity
v = the original velocity (0.50c in this case)
u = the velocity being added (0.50c in this case)
c = the speed of light

Let's calculate the spaceship's speed after each increment:

1st increment:
v' = (0.50c + 0.50c) / (1 + (0.50c * 0.50c) / (c^2))
= c / (1 + 0.25)
= c / 1.25
= 0.8c

2nd increment:
v' = (0.8c + 0.50c) / (1 + (0.8c * 0.50c) / (c^2))
= 1.30c / (1 + 0.40)
= 1.30c / 1.40
= 0.9286c

3rd increment:
v' = (0.9286c + 0.50c) / (1 + (0.9286c * 0.50c) / (c^2))
= 1.4286c / (1 + 0.145c)
= 1.4286c / 1.145c
= 1.25c

4th increment:
v' = (1.25c + 0.50c) / (1 + (1.25c * 0.50c) / (c^2))
= 1.75c / (1 + 0.3125)
= 1.75c / 1.3125
= 1.3333c

5th increment:
v' = (1.3333c + 0.50c) / (1 + (1.3333c * 0.50c) / (c^2))
= 1.8333c / (1 + 0.4167)
= 1.8333c / 1.4167
= 1.2941c

6th increment:
v' = (1.2941c + 0.50c) / (1 + (1.2941c * 0.50c) / (c^2))
= 1.7941c / (1 + 0.3235)
= 1.7941c / 1.3235
= 1.3559c

7th increment:
v' = (1.3559c + 0.50c) / (1 + (1.3559c * 0.50c) / (c^2))
= 1.8559c / (1 + 0.338)
= 1.8559c / 1.338
= 1.3897c

As we can see, the spaceship's speed exceeds 0.999c after the 7th increment, reaching approximately 1.3897c.

Therefore, it requires 7 increments for the spaceship's speed to exceed 0.999c.