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 can follow these steps:

### Step-by-Step Solution:

1. **Understand the Path**:
The body moves along a quarter circle from point A to point B. This means the path taken is a quarter of the circumference of a circle.

2. **Calculate the Distance**:
The distance traveled along the quarter circle can be calculated using the formula for the circumference of a circle, which is \(C = 2\pi R\). For a quarter circle, the distance \(D\) is:
\[ D = \frac{1}{4} \times 2\pi R = \frac{\pi R}{2} \]
(Note: The video transcript incorrectly used \(22/7\) instead of \(\pi\). For the sake of this solution, we will use \(\pi\) for accuracy.)

3. **Calculate the Displacement**:
The displacement is the straight line distance from point A to point B. In this case, since it forms a right triangle (with both legs equal to the radius \(R\)), we can use the Pythagorean theorem:
\[ S = \sqrt{R^2 + R^2} = \sqrt{2R^2} = R\sqrt{2} \]

4. **Calculate the Ratio of Distance to Displacement**:
Now we can find the ratio of distance to displacement:
\[ \text{Ratio} = \frac{D}{S} = \frac{\frac{\pi R}{2}}{R\sqrt{2}} = \frac{\pi}{2\sqrt{2}} \]

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

### Final Result:
The ratio of distance to displacement is:
\[ \frac{\pi \sqrt{2}}{4} \]