A body moves along a curved path of a quarter circle. Calculate the ratio of distance to displacement :

Options

A. `11:7`
B. `7:11`
C. `11: sqrt(2) xx7`
D. `7:11 sqrt(2)`

Answer

To solve the problem of finding the ratio of distance to displacement for a body moving along a curved path of a quarter circle, we will follow these steps:

### Step 1: Understand the Path
The body moves along a quarter circle. This means that if we consider a circle with radius \( r \), the body travels along a path that is one-fourth of the circumference of that circle.

### Step 2: Calculate the Distance
The formula for the circumference of a circle is given by:
\[ C = 2\pi r \]
Since the body moves along a quarter of the circle, the distance \( d \) traveled 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 defined as the shortest straight-line distance from the initial position to the final position. For a quarter circle, the initial position is at one end of the radius and the final position is at the other end of the radius. The displacement \( s \) 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 distance and displacement, we can find the ratio:
\[ \text{Ratio} = \frac{\text{Distance}}{\text{Displacement}} = \frac{\frac{\pi r}{2}}{r\sqrt{2}} = \frac{\pi}{2\sqrt{2}} \]

### Step 5: Simplify the Ratio
To express the ratio in a simpler form, we can multiply the numerator and denominator by \(\sqrt{2}\):
\[ \text{Ratio} = \frac{\pi \sqrt{2}}{2 \cdot 2} = \frac{\pi \sqrt{2}}{4} \]

### Final Answer
Thus, the ratio of distance to displacement is:
\[ \frac{\pi \sqrt{2}}{4} \]