A star is moving away from earth and shift in spectral line of wavelength `5700 Å` is `1.90 Å`. Velocity of the star is :

Options

A. (a)`50kms^(-1)`
B. (b)`70kms^(-1)`
C. (c)`85kms^(-1)`
D. (d)`100kms^(-1)`

Answer

To solve the problem of finding the velocity of a star that is moving away from Earth, we can use the Doppler effect formula for light. The steps to solve the problem are as follows:

### Step-by-Step Solution:

1. **Identify the given values:**
- Wavelength of the spectral line, \( \lambda = 5700 \, \text{Å} \)
- Shift in wavelength, \( \Delta \lambda = 1.90 \, \text{Å} \)

2. **Convert the wavelengths from Angstroms to meters:**
- \( 1 \, \text{Å} = 10^{-10} \, \text{m} \)
- Therefore,
\[ \lambda = 5700 \, \text{Å} = 5700 \times 10^{-10} \, \text{m} = 5.7 \times 10^{-7} \, \text{m} \]
- And,
\[ \Delta \lambda = 1.90 \, \text{Å} = 1.90 \times 10^{-10} \, \text{m} \]

3. **Use the Doppler effect formula:**
The formula relating the change in wavelength to the velocity of the star is given by:
\[ \frac{\Delta \lambda}{\lambda} = \frac{v}{c} \]
where \( v \) is the velocity of the star and \( c \) is the speed of light (\( c \approx 3 \times 10^8 \, \text{m/s} \)).

4. **Rearrange the formula to solve for \( v \):**
\[ v = c \cdot \frac{\Delta \lambda}{\lambda} \]

5. **Substitute the known values into the equation:**
\[ v = 3 \times 10^8 \, \text{m/s} \cdot \frac{1.90 \times 10^{-10} \, \text{m}}{5.7 \times 10^{-7} \, \text{m}} \]

6. **Calculate the fraction:**
\[ \frac{1.90 \times 10^{-10}}{5.7 \times 10^{-7}} \approx 0.000332 \]

7. **Calculate the velocity:**
\[ v \approx 3 \times 10^8 \, \text{m/s} \cdot 0.000332 \approx 99600 \, \text{m/s} \]

8. **Convert the velocity to kilometers per second:**
\[ v \approx 99.6 \, \text{km/s} \]

9. **Round to the nearest whole number:**
\[ v \approx 100 \, \text{km/s} \]

10. **Conclusion:**
The velocity of the star moving away from Earth is approximately \( 100 \, \text{km/s} \).

### Final Answer:
The velocity of the star is \( 100 \, \text{km/s} \).