The wavelength of spectral line coming from a distant star shifts from 600 nm to 600.1 nm. The velocity of the star relative to earth is

Options

A. 50 km `s^(-1)`
B. 100 km `s^(-1)`
C. 25 km `s^(-1)`
D. 200 km `s^(-1)`

Explanation

To find the velocity of the star relative to Earth based on the shift in the wavelength of a spectral line, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Wavelengths**:
- The original wavelength (\(\lambda_1\)) is 600 nm.
- The shifted wavelength (\(\lambda_2\)) is 600.1 nm.

2. **Calculate the Change in Wavelength**:
- The change in wavelength (\(\Delta \lambda\)) can be calculated as:
\[ \Delta \lambda = \lambda_2 - \lambda_1 = 600.1 \, \text{nm} - 600 \, \text{nm} = 0.1 \, \text{nm} \]

3. **Use the Formula for Relative Velocity**:
- The formula to calculate the relative velocity (\(v\)) of the star is given by:
\[ v = \frac{\Delta \lambda}{\lambda_1} \times c \]
- Where \(c\) is the speed of light, approximately \(3 \times 10^8 \, \text{m/s}\).

4. **Substitute the Values into the Formula**:
- Substituting the values we have:
\[ v = \frac{0.1 \, \text{nm}}{600 \, \text{nm}} \times 3 \times 10^8 \, \text{m/s} \]

5. **Simplify the Expression**:
- First, calculate the fraction:
\[ \frac{0.1}{600} = \frac{1}{6000} \]
- Now substitute this back into the equation:
\[ v = \frac{1}{6000} \times 3 \times 10^8 \, \text{m/s} \]

6. **Calculate the Velocity**:
- Calculate \(v\):
\[ v = \frac{3 \times 10^8}{6000} \, \text{m/s} = 5 \times 10^4 \, \text{m/s} \]

7. **Convert to Kilometers per Second**:
- To convert from meters per second to kilometers per second, divide by 1000:
\[ v = \frac{5 \times 10^4 \, \text{m/s}}{1000} = 50 \, \text{km/s} \]

### Final Answer:
The velocity of the star relative to Earth is **50 km/s**.

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