The spectral line of a given element in the light received from a distant star is shifted towards the longer wavelength by `0.032%`. Duduce the velocity of star in the line of sight.

Options

A. `96 km"/"sec`
B. `64 km"/"sec`
C. `115 km"/"sec`
D. `30 km"/"sec`

Answer

To solve the problem of determining the velocity of a star based on the shift in the spectral line, we can follow these steps:

### Step 1: Understand the given information
The problem states that the spectral line is shifted towards longer wavelengths by 0.032%. This indicates a redshift, which occurs when an object is moving away from the observer.

### Step 2: Convert the percentage shift to a decimal
The percentage shift given is 0.032%. To use this in calculations, we need to convert it to a decimal form:
\[ \text{Shift} = \frac{0.032}{100} = 0.00032 \]

### Step 3: Relate the shift to the velocity
The relationship between the change in wavelength (\(\Delta \lambda\)) and the velocity (\(v\)) of the star can be expressed using the formula:
\[ \frac{\Delta \lambda}{\lambda} = \frac{v}{c} \]
where \(c\) is the speed of light, approximately \(3 \times 10^8 \, \text{m/s}\).

### Step 4: Rearranging the formula to solve for velocity
From the above relationship, we can rearrange it to find the velocity:
\[ v = c \cdot \frac{\Delta \lambda}{\lambda} \]

### Step 5: Substitute the values into the equation
We substitute the decimal shift we calculated and the speed of light into the equation:
\[ v = 3 \times 10^8 \, \text{m/s} \cdot 0.00032 \]

### Step 6: Calculate the velocity
Now we perform the calculation:
\[ v = 3 \times 10^8 \cdot 0.00032 = 96000 \, \text{m/s} \]

### Step 7: Convert the velocity to kilometers per second
To convert from meters per second to kilometers per second, we divide by 1000:
\[ v = \frac{96000}{1000} = 96 \, \text{km/s} \]

### Conclusion
Thus, the velocity of the star in the line of sight is \(96 \, \text{km/s}\).