The wavelength of light coming from a distant galaxy is found to be `0.5%` more than that coming from a source on earth. Calculate the velocity of galaxy.

Options

A. `3xx10^(10)ms^-1`
B. `1.5xx10^(10)ms^-1`
C. `1.5xx10^(8)ms^-1`
D. `1.5xx10^(6)ms^-1`

Answer

To solve the problem, we need to calculate the velocity of a distant galaxy based on the change in wavelength of light emitted from it. Here's the step-by-step solution:

### Step 1: Understand the Change in Wavelength
The problem states that the wavelength of light from the galaxy is `0.5%` more than that from a source on Earth. This means we can express the change in wavelength as a fractional change.

### Step 2: Calculate the Fractional Change in Wavelength
The fractional change in wavelength (Δλ/λ) can be calculated as:
\[ \frac{\Delta \lambda}{\lambda} = \frac{0.5}{100} = 0.005 \]

### Step 3: Relate the Change in Wavelength to Velocity
According to the Doppler effect for light, the change in wavelength is related to the velocity of the source (in this case, the galaxy) as follows:
\[ \frac{\Delta \lambda}{\lambda} = \frac{v}{c} \]
where \( v \) is the velocity of the galaxy and \( c \) is the speed of light (approximately \( 3 \times 10^8 \) m/s).

### Step 4: Rearranging the Formula to Find Velocity
We can rearrange the formula to solve for the velocity \( v \):
\[ v = c \cdot \frac{\Delta \lambda}{\lambda} \]

### Step 5: Substitute the Values
Substituting the known values into the equation:
\[ v = 3 \times 10^8 \, \text{m/s} \times 0.005 \]

### Step 6: Calculate the Velocity
Now, we perform the multiplication:
\[ v = 3 \times 10^8 \times 0.005 = 1.5 \times 10^6 \, \text{m/s} \]

### Final Answer
The velocity of the galaxy is \( 1.5 \times 10^6 \, \text{m/s} \).

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