The spectral line of wavelength `lambda=5000 Å` in the light coming from a distant star is observed as 5200 Å.Determine the recession velocity of the star.

Answer

To determine the recession velocity of the star, we can use the formula derived from the Doppler effect for light, which relates the change in wavelength to the velocity of the source of light. Here’s the step-by-step solution:

### Step 1: Identify the given values
- Observed wavelength (\( \lambda' \)) = 5200 Å
- Emitted wavelength (\( \lambda \)) = 5000 Å

### Step 2: Calculate the change in wavelength
The change in wavelength (\( \Delta \lambda \)) can be calculated as:
\[ \Delta \lambda = \lambda' - \lambda \]
Substituting the values:
\[ \Delta \lambda = 5200 \, \text{Å} - 5000 \, \text{Å} = 200 \, \text{Å} \]

### Step 3: Use the formula for recession velocity
The recession velocity (\( v \)) can be calculated using the formula:
\[ v = \frac{\Delta \lambda}{\lambda} \cdot c \]
where \( c \) is the speed of light (\( c \approx 3 \times 10^8 \, \text{m/s} \)).

### Step 4: Substitute the values into the formula
First, convert the wavelengths from Ångströms to meters:
\[ \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \]
\[ \Delta \lambda = 200 \, \text{Å} = 200 \times 10^{-10} \, \text{m} = 2 \times 10^{-8} \, \text{m} \]

Now, substitute the values into the recession velocity formula:
\[ v = \frac{2 \times 10^{-8} \, \text{m}}{5 \times 10^{-7} \, \text{m}} \cdot (3 \times 10^8 \, \text{m/s}) \]

### Step 5: Calculate the fraction
\[ \frac{2 \times 10^{-8}}{5 \times 10^{-7}} = \frac{2}{5} \times 10^{(-8 + 7)} = \frac{2}{5} \times 10^{-1} = 0.4 \]

### Step 6: Calculate the recession velocity
Now, multiply by the speed of light:
\[ v = 0.4 \cdot (3 \times 10^8 \, \text{m/s}) = 1.2 \times 10^8 \, \text{m/s} \]

### Final Answer
The recession velocity of the star is \( v = 1.2 \times 10^8 \, \text{m/s} \).

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