A body covered a distance of `L m` along a curved path of a quarter circle. The ratio of distance to displacements.

Options

A. `(pi)/(2 sqrt(2))`
B. `(2 sqrt(2))/(pi)`
C. `(pi)/(sqrt(2))`
D. `(sqrt(2))/(pi)`

Explanation

To solve the problem of finding the ratio of distance to displacement for a body that covers a distance of \( L \) meters along a curved path of a quarter circle, we can follow these steps:

### Step 1: Understand the Geometry of the Problem
A quarter circle is a segment of a circle that represents 90 degrees of the full circle. If we denote the radius of the quarter circle as \( r \), then the distance traveled along the curved path (the arc length) can be calculated.

### Step 2: Calculate the Distance
The distance \( d \) covered along the quarter circle can be calculated using the formula for the circumference of a circle:
\[ \text{Circumference of full circle} = 2\pi r \]
Since we are only considering a quarter circle, the distance \( d \) is:
\[ d = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \]

### Step 3: Calculate the Displacement
The displacement \( s \) is the straight-line distance between the starting point and the ending point of the quarter circle. In this case, the starting point \( A \) is at one end of the radius, and the ending point \( B \) is at the other end of the radius. The displacement can be calculated using the Pythagorean theorem:
\[ s = \sqrt{(r)^2 + (r)^2} = \sqrt{2r^2} = r\sqrt{2} \]

### Step 4: Find the Ratio of Distance to Displacement
Now that we have both the distance and the displacement, we can find the ratio:
\[ \text{Ratio} = \frac{d}{s} = \frac{\frac{\pi r}{2}}{r\sqrt{2}} \]
Simplifying this expression, we get:
\[ \text{Ratio} = \frac{\pi r}{2r\sqrt{2}} = \frac{\pi}{2\sqrt{2}} \]

### Final Answer
Thus, the ratio of distance to displacement is:
\[ \frac{\pi}{2\sqrt{2}} \]