A car is going round a circle of radius `R_(1)` with constant speed. Another car is going round a circle of radius `R_(2)` with constant speed. If both of them take same time to complete the circles, the ratio of their angular speeds and linear speeds will be

Options

A. `sqrt((R_(1))/(R_(2))),(R_(1))/(R_(2))`
B. 1,1
C. 1, `(R_(1))/(R_(2))`
D. `(R_(1))/(R_(2))` , 1

Answer

To solve the problem, we need to find the ratios of angular speeds and linear speeds of two cars moving in circles of different radii but taking the same time to complete their respective circles.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two cars: Car 1 moving in a circle of radius \( R_1 \) and Car 2 moving in a circle of radius \( R_2 \).
- Both cars take the same time \( T \) to complete one full revolution.

2. **Calculating Linear Speed**:
- The linear speed \( V \) of an object moving in a circle is given by the formula:
\[ V = \frac{\text{Distance}}{\text{Time}} \]
- The distance traveled in one complete revolution (circumference) for Car 1 is:
\[ V_1 = \frac{2\pi R_1}{T} \]
- Similarly, for Car 2:
\[ V_2 = \frac{2\pi R_2}{T} \]

3. **Calculating Angular Speed**:
- The angular speed \( \omega \) is defined as:
\[ \omega = \frac{\text{Angle}}{\text{Time}} \]
- For one complete revolution, the angle is \( 2\pi \) radians. Thus, for Car 1:
\[ \omega_1 = \frac{2\pi}{T} \]
- For Car 2:
\[ \omega_2 = \frac{2\pi}{T} \]

4. **Finding the Ratio of Angular Speeds**:
- The ratio of angular speeds \( \frac{\omega_1}{\omega_2} \) is:
\[ \frac{\omega_1}{\omega_2} = \frac{\frac{2\pi}{T}}{\frac{2\pi}{T}} = 1 \]

5. **Finding the Ratio of Linear Speeds**:
- The ratio of linear speeds \( \frac{V_1}{V_2} \) is:
\[ \frac{V_1}{V_2} = \frac{\frac{2\pi R_1}{T}}{\frac{2\pi R_2}{T}} = \frac{R_1}{R_2} \]

6. **Final Result**:
- The ratio of angular speeds is \( 1:1 \) and the ratio of linear speeds is \( R_1:R_2 \).

### Conclusion:
The final ratios are:
- Ratio of angular speeds: \( 1:1 \)
- Ratio of linear speeds: \( R_1:R_2 \)