A body completes one round of a circle of radius `R` in 20 second. The displacement of the body after 45 seconds is

Options

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

Explanation

To solve the problem step by step, we will analyze the motion of the body in a circular path and calculate the displacement after 45 seconds.

### Step 1: Determine the time for one complete revolution
The body completes one full revolution around the circle in 20 seconds.

### Step 2: Calculate the number of complete revolutions in 45 seconds
To find out how many complete revolutions the body makes in 45 seconds, we can divide the total time by the time for one revolution:
\[ \text{Number of complete revolutions} = \frac{45 \text{ seconds}}{20 \text{ seconds/revolution}} = 2.25 \text{ revolutions} \]

### Step 3: Determine the position after 2 complete revolutions
After 2 complete revolutions (which takes 40 seconds), the body returns to its starting point.

### Step 4: Calculate the additional time after 40 seconds
After 40 seconds, there are 5 seconds remaining (45 seconds - 40 seconds = 5 seconds).

### Step 5: Determine the fraction of the revolution completed in the remaining 5 seconds
In 20 seconds, the body completes 1 full revolution. Therefore, in 5 seconds, the fraction of the revolution completed is:
\[ \text{Fraction of revolution in 5 seconds} = \frac{5 \text{ seconds}}{20 \text{ seconds}} = \frac{1}{4} \]

### Step 6: Determine the final position after 45 seconds
Starting from the initial position (0 degrees), after 2 complete revolutions, the body is back at the starting point (0 degrees). After an additional \( \frac{1}{4} \) of a revolution, the body moves to:
\[ \text{Final position} = 0 + 90^\circ = 90^\circ \]

### Step 7: Calculate the displacement
The displacement is the straight line distance from the initial position to the final position. Since the radius of the circle is \( R \), the coordinates of the initial position (0 degrees) are (R, 0) and the coordinates of the final position (90 degrees) are (0, R).

To find the displacement, we can use the distance formula:
\[ \text{Displacement} = \sqrt{(0 - R)^2 + (R - 0)^2} = \sqrt{R^2 + R^2} = \sqrt{2R^2} = R\sqrt{2} \]

### Final Answer
The displacement of the body after 45 seconds is:
\[ \text{Displacement} = R\sqrt{2} \]