Answer:
the velocity of the spaceship relative to the first planet is approximately 299,792,457.7 meters per second.
Step-by-step explanation:
To determine the velocity of the spaceship relative to the first planet, we can use the relativistic velocity addition formula. The formula is as follows:
v' = (v1 + v2) / (1 + (v1*v2) / c^2)
Where:
v' is the relative velocity between the two planets
v1 is the velocity of the spaceship as observed by the first planet (0.750c)
v2 is the velocity of the two planets approaching each other (0.250c)
c is the speed of light in a vacuum
Plugging in the given values, we can calculate the relative velocity:
v' = (0.750c + 0.250c) / (1 + (0.750c * 0.250c) / c^2)
= (1.000c) / (1 + (0.1875) / c^2)
Since c represents the speed of light, which is approximately 299,792,458 meters per second, we can substitute it into the equation:
v' = (1.000 * 299,792,458 m/s) / (1 + (0.1875) / (299,792,458 m/s)^2)
≈ 299,792,458 m/s / (1 + 5.917 * 10^(-10))
≈ 299,792,458 m/s / (1.0000000005917)
Thus, the velocity of the spaceship relative to the first planet is approximately 299,792,457.7 meters per second.