Final answer:
The calculated relative velocity of the spaceship to the first planet exceeds the speed of light, which indicates an error in the problem as the correct answer cannot exceed the speed of light (c).
Step-by-step explanation:
The student has asked about the velocity of a spaceship relative to the first planet when two planets are heading towards each other and a spaceship is sent from one planet to another. To solve this, we need to understand the concepts of relative velocity in the context of special relativity, as the velocities involved are a significant fraction of the speed of light (represented by c).
We are given that the two planets are approaching each other at 0.250c, and the spaceship approaches the second planet at 0.750c as observed by the second planet. The velocity of the spaceship relative to the first planet can be found by using the formula for adding velocities in relativity:
Vtotal = (V1 + V2) / (1 + (V1 ⋅ V2 / c2))
Here, V1 is the velocity of the second planet relative to the first, which is 0.250c, and V2 is the velocity of the spaceship relative to the second planet, which is 0.750c. Plugging the values in, we get:
Vtotal = (0.250c + 0.750c) / (1 + (0.250c ⋅ 0.750c / c2))
Vtotal = 1.000c / (1 + 0.1875c2 / c2) = 1.000c / (1 + 0.1875) = 1.000c / 1.1875 = 0.842c
However, this result exceeds the speed of light, which is not possible according to the laws of physics. The problem seems to be ill-posed or contain incorrect information since the velocity of the ship relative to the planets exceeds the speed of light. In reality, the relative velocity would never reach or surpass c.