The wavelength of light of a particular wavelength received from a galaxy is measured on earth and is found to be 5% more that its wavelength. It follows that the galaxy is:

Options

A. Approaching the earth with a speed `3xx 10^(7) ms^(-1)`
B. Going away from the earth with a speed `1.5 xx 10^(7) ms^(-1)`
C. Approaching the earth with a speed `1.5 xx 10^(7) ms^(-1)`
D. Going away from the earth with a speed `1.5xx10^(7) ms^(-1)`

Explanation

To solve the problem, we need to analyze the situation using the concept of redshift, which occurs when the wavelength of light from an object moving away from an observer is increased. Here are the steps to arrive at the solution:

### Step-by-Step Solution:

1. **Understanding the Shift in Wavelength**:
- We are given that the observed wavelength (\( \lambda' \)) is 5% more than the original wavelength (\( \lambda_0 \)). This can be expressed as:
\[ \lambda' = \lambda_0 + 0.05 \lambda_0 = 1.05 \lambda_0 \]

2. **Using the Redshift Formula**:
- The relationship between the observed wavelength and the original wavelength due to the Doppler effect when the source is moving away is given by:
\[ \lambda' = \lambda_0 \left(1 + \frac{v}{c}\right) \]
- Here, \( v \) is the velocity of the galaxy, and \( c \) is the speed of light.

3. **Setting Up the Equation**:
- From the above relationship, we can substitute \( \lambda' \):
\[ 1.05 \lambda_0 = \lambda_0 \left(1 + \frac{v}{c}\right) \]

4. **Cancelling \( \lambda_0 \)**:
- Since \( \lambda_0 \) is common on both sides, we can cancel it (assuming \( \lambda_0 \neq 0 \)):
\[ 1.05 = 1 + \frac{v}{c} \]

5. **Solving for Velocity \( v \)**:
- Rearranging the equation gives:
\[ \frac{v}{c} = 1.05 - 1 = 0.05 \]
- Now, multiplying both sides by \( c \):
\[ v = 0.05c \]

6. **Substituting the Speed of Light**:
- The speed of light \( c \) is approximately \( 3 \times 10^8 \) m/s. Thus:
\[ v = 0.05 \times 3 \times 10^8 = 1.5 \times 10^7 \text{ m/s} \]

7. **Conclusion**:
- The galaxy is moving away from the Earth with a speed of \( 1.5 \times 10^7 \) m/s.

### Final Answer:
The galaxy is moving away from the Earth with a speed of \( 1.5 \times 10^7 \) m/s.

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