The spectrum of distant nebula is taken in an observatory. A shift of 0.025% is observed in the spectral lines. Calculate the velocity of the distant nebula.

Answer

To solve the problem of calculating the velocity of a distant nebula based on the observed shift in its spectral lines, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Shift in Wavelength**:
The problem states that there is a shift of 0.025% in the spectral lines. This shift can be expressed in terms of the natural wavelength (λ) of the light emitted by the nebula.

2. **Calculate the Shift in Wavelength (Δλ)**:
The shift in wavelength (Δλ) can be calculated as:
\[ \Delta \lambda = \frac{0.025}{100} \times \lambda = 0.00025 \lambda \]

3. **Use the Doppler Effect Formula**:
The relationship between the shift in wavelength and the velocity (v) of the nebula is given by the formula:
\[ \frac{\Delta \lambda}{\lambda} = \frac{v}{c} \]
where \( c \) is the speed of light (approximately \( 3 \times 10^8 \) m/s).

4. **Substitute Δλ into the Formula**:
Substitute the expression for Δλ into the Doppler effect formula:
\[ \frac{0.00025 \lambda}{\lambda} = \frac{v}{c} \]
This simplifies to:
\[ 0.00025 = \frac{v}{c} \]

5. **Solve for Velocity (v)**:
Rearranging the equation gives:
\[ v = 0.00025 \times c \]
Now substitute the value of \( c \):
\[ v = 0.00025 \times 3 \times 10^8 \text{ m/s} \]

6. **Calculate the Velocity**:
Performing the multiplication:
\[ v = 0.00025 \times 3 = 0.00075 \]
Then, multiplying by \( 10^8 \):
\[ v = 0.00075 \times 10^8 = 7.5 \times 10^4 \text{ m/s} \]

7. **Final Result**:
The velocity of the distant nebula is:
\[ v = 7.5 \times 10^4 \text{ m/s} \]

### Final Answer:
The velocity of the distant nebula is \( 7.5 \times 10^4 \) m/s.