A man of height 1.47 m stands on a straight road on a hot day. The vertical temperature in the air results in a variation of refractive index with height y as `mu=mu_(0)sqrt((1+ay))` where `mu_(0)` is the refractive index of air near the road a `=1.5xx10^(-6)`/m. What is the apparent length of the road man is able to sec?

Options

A. 700m
B. 2000 m
C. `7000sqrt(2)`
D. infinite distance

Answer

To solve the problem, we need to determine the apparent length of the road that the man can see, given the variation of the refractive index with height. Let's break down the solution step by step.

### Step 1: Understand the refractive index variation
The refractive index \( \mu \) varies with height \( y \) as:
\[ \mu = \mu_0 \sqrt{1 + ay} \]
where \( \mu_0 \) is the refractive index of air near the road, and \( a = 1.5 \times 10^{-6} \, \text{m}^{-1} \).

### Step 2: Determine the height of the man
The height of the man is given as \( h = 1.47 \, \text{m} \).

### Step 3: Use the small angle approximation
For small angles, the apparent distance \( x \) that the man can see can be related to the height \( h \) and the refractive index. The relationship can be derived from the geometry of the situation and the behavior of light in a medium with varying refractive index.

### Step 4: Set up the relationship
Using the formula for the apparent distance:
\[ \frac{dy}{dx} = \frac{\sqrt{\mu^2 - \sin^2 \theta}}{\mu \sin \theta} \]
where \( \mu = \mu_0 \sqrt{1 + ay} \).

### Step 5: Substitute the values
We can simplify this to find \( dx \) in terms of \( dy \):
\[ \frac{dy}{dx} = \sqrt{\mu_0^2 (1 + ay) - \sin^2 \theta} \div (\mu_0 \sin \theta) \]

### Step 6: Integrate to find the total distance
We integrate from \( y = 0 \) to \( y = h \) (1.47 m) to find the total distance \( x \):
\[ x = \int_0^{h} \frac{dy}{\sqrt{\mu_0^2 (1 + ay) - \sin^2 \theta}} \]

### Step 7: Solve for the apparent length
After performing the integration and substituting the values, we find:
\[ x = \frac{2 \sqrt{h}}{\sqrt{\mu_0 a}} \]

### Step 8: Substitute known values
Assuming \( \mu_0 \approx 1 \) (the refractive index of air), we have:
\[ x = \frac{2 \sqrt{1.47}}{\sqrt{1 \cdot 1.5 \times 10^{-6}}} \]

Calculating this gives:
\[ x \approx 1979 \, \text{m} \]

### Step 9: Final result
Thus, the apparent length of the road that the man can see is approximately \( 2000 \, \text{m} \).