An astronaut aged 3 years when traveling at 99% of the speed of light to the Andromeda galaxy and back. The space officials who greeted her on her return have aged less than

Options

A. 3 years
B. More than 2.5 years and less than 3 years
C. More then 3 years
D. 2.5 years

Answer

To solve the problem of how much the space officials aged while the astronaut traveled to the Andromeda galaxy and back at 99% of the speed of light, we will use the concept of time dilation from the theory of relativity.

### Step-by-Step Solution:

1. **Understanding Time Dilation**:
In special relativity, time dilation occurs when an object moves at a significant fraction of the speed of light. The time experienced by the moving observer (the astronaut) is less than the time experienced by a stationary observer (the space officials).

2. **Identify the Speed**:
The speed of the astronaut is given as 99% of the speed of light (v = 0.99c).

3. **Calculate the Lorentz Factor (γ)**:
The Lorentz factor (γ) is calculated using the formula:
\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]
Substituting v = 0.99c:
\[ \gamma = \frac{1}{\sqrt{1 - (0.99)^2}} = \frac{1}{\sqrt{1 - 0.9801}} = \frac{1}{\sqrt{0.0199}} \approx 7.09 \]

4. **Determine the Time Experienced by the Astronaut**:
The astronaut aged 3 years during the trip. This is the proper time (Δt') experienced by the astronaut.

5. **Calculate the Time Experienced by the Space Officials**:
The time experienced by the space officials (Δt) can be calculated using the time dilation formula:
\[ \Delta t = \gamma \Delta t' \]
Substituting the values:
\[ \Delta t = 7.09 \times 3 \text{ years} \approx 21.27 \text{ years} \]

6. **Conclusion**:
The space officials aged approximately 21.27 years while the astronaut aged 3 years. Therefore, the space officials aged significantly more than the astronaut.

### Final Answer:
The space officials aged approximately 21.27 years.

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