An object covers 1/4 of the circular path, what will be the ratio of the distance and displacement of the object ?

Options

A. `(pi)/(2sqrt(2))`
B. `(pi)/(2sqrt(3))`
C. `(pi)/(2sqrt(12))`
D. `(pi)/(3sqrt(2))`

Explanation

To solve the problem of finding the ratio of distance to displacement when an object covers 1/4 of a circular path, we can follow these steps:

### Step 1: Understand the Circular Path
The object moves along a circular path. When it covers 1/4 of the circular path, it essentially moves along a quarter of the circumference of the circle.

### Step 2: Calculate the Distance Covered
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. For 1/4 of the circular path, the distance \( D \) covered is:
\[ D = \frac{1}{4} \times C = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \]

### Step 3: Calculate the Displacement
Displacement is the shortest distance between the initial and final positions of the object. When the object moves 1/4 of the circular path, it moves from one point on the circle to another point that is 90 degrees away. The displacement \( S \) can be calculated using the Pythagorean theorem, as the two points form a right triangle with the radius as the legs:
\[ S = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2} \]

### Step 4: Find the Ratio of Distance to Displacement
Now, we can find the ratio of distance \( D \) to displacement \( S \):
\[ \text{Ratio} = \frac{D}{S} = \frac{\frac{\pi r}{2}}{r\sqrt{2}} \]
The \( r \) cancels out:
\[ \text{Ratio} = \frac{\pi}{2\sqrt{2}} \]

### Final Answer
Thus, the ratio of the distance to the displacement when an object covers 1/4 of a circular path is:
\[ \frac{\pi}{2\sqrt{2}} \]

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