Final answer:
Using the formula for relativistic addition of velocities, the velocity of the canister relative to the Earth is found to be approximately 0.909c. However, none of the provided options match this result, indicating a possible error in the question.
Step-by-step explanation:
The situation described involves the concept of relativistic addition of velocities, which is a principle from special relativity that applies to objects moving at significant fractions of the speed of light (denoted as c). When adding velocities in a relativistic context, one cannot simply add them as in classical mechanics due to the consequences of Einstein's theory of relativity.
According to the formula for the relativistic addition of velocities:
u = (u' + v) / (1 + (u'v/c²))
where:
- u is the velocity of the canister relative to the Earth
- u' is the velocity of the canister relative to the spaceship, 0.500c
- v is the velocity of the spaceship relative to Earth, 0.750c
Plugging in the given values:
u = (0.500c + 0.750c) / (1 + (0.500c * 0.750c / c²))
Thus the velocity of the canister relative to the Earth if it is shot directly at the Earth is:
u = (1.250c) / (1 + 0.375)
u = 1.250c / 1.375
u ≈ 0.909c
However, this answer is not one of the provided options, which suggests a possible typo or error in the question. Based on the principle of relativistic addition of velocities and the provided options, the answer should be (b) 1.250c, but this exceeds the speed of light, which is not possible according to special relativity.