Spaceship A travels at 0.400c relative to an Earth observer. According to the same observer, spaceship A overtakes a slower moving spaceship B that moves in the same direction. The captain of B sees A pass her ship at 0.114c. Determine the speed of B relative to the Earth observer.

Options

A. 0.100c
B. 0.300c
C. 0.214c
D. 0.625c

Answer

To determine the speed of spaceship B relative to the Earth observer, we will use the relativistic velocity addition formula. The formula is given by:

\[ v' = \frac{u + v}{1 + \frac{uv}{c^2}} \]

where:
- \( v' \) is the speed of spaceship B relative to the Earth observer,
- \( u \) is the speed of spaceship A relative to the Earth observer,
- \( v \) is the speed of spaceship A relative to spaceship B,
- \( c \) is the speed of light.

### Step-by-Step Solution:

1. **Identify the given values**:
- Speed of spaceship A relative to the Earth observer, \( u = 0.400c \)
- Speed of spaceship A relative to spaceship B, \( v = 0.114c \)

2. **Substitute the values into the relativistic velocity addition formula**:
We need to find \( v' \) (the speed of spaceship B relative to the Earth observer):
\[ v' = \frac{u - v}{1 - \frac{uv}{c^2}} \]

Here, we use \( -v \) because spaceship B is moving in the same direction as spaceship A.

3. **Calculate the numerator**:
\[ u - v = 0.400c - 0.114c = 0.286c \]

4. **Calculate the denominator**:
\[ 1 - \frac{uv}{c^2} = 1 - \frac{(0.400)(0.114)}{1} = 1 - 0.0456 = 0.9544 \]

5. **Combine the results**:
\[ v' = \frac{0.286c}{0.9544} \approx 0.299c \]

6. **Final result**:
The speed of spaceship B relative to the Earth observer is approximately:
\[ v' \approx 0.300c \]