A telescope of aperture diameter 5m is used to observe the moon from the earth. Distance between the moon and earth is `4 xx 10^5` km. Determine the minimum distance between two points on the moon's surface which can be resolved using this telescope. (Wave length of light is `5893 A^@`.

Options

A. 60m
B. 20 m
C. 600 m
D. 200 m

Explanation

To determine the minimum distance between two points on the moon's surface that can be resolved using a telescope, we can use the formula for angular resolution given by Rayleigh's criterion:

\[ \theta = \frac{1.22 \lambda}{D} \]

Where:
- \(\theta\) is the angular resolution in radians,
- \(\lambda\) is the wavelength of light,
- \(D\) is the diameter of the telescope's aperture.

### Step 1: Convert the given values to appropriate units
- The wavelength \(\lambda = 5893 \, \text{Å} = 5893 \times 10^{-10} \, \text{m}\)
- The diameter of the telescope's aperture \(D = 5 \, \text{m}\)
- The distance from the Earth to the Moon \(r = 4 \times 10^5 \, \text{km} = 4 \times 10^8 \, \text{m}\)

### Step 2: Calculate the angular resolution \(\theta\)
Using the formula:

\[ \theta = \frac{1.22 \lambda}{D} \]

Substituting the values:

\[ \theta = \frac{1.22 \times 5893 \times 10^{-10}}{5} \]

Calculating this gives:

\[ \theta \approx \frac{1.22 \times 5893 \times 10^{-10}}{5} \approx 1.44 \times 10^{-13} \, \text{radians} \]

### Step 3: Calculate the minimum resolvable distance on the moon's surface
The minimum distance \(d\) between two points on the moon's surface that can be resolved is given by:

\[ d = r \cdot \theta \]

Substituting the values:

\[ d = (4 \times 10^8) \cdot (1.44 \times 10^{-13}) \]

Calculating this gives:

\[ d \approx 57.6 \, \text{m} \]

### Step 4: Round to the nearest option
Since the closest option to our calculated value is 60 m, we can conclude that:

\[ \text{Minimum distance between two points on the moon's surface} \approx 60 \, \text{m} \]

### Final Answer
The minimum distance between two points on the moon's surface which can be resolved using this telescope is approximately **60 meters**.

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