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 circular paths of different radii but taking the same time to complete their respective circles.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the radius of the first circle be \( R_1 \).
- Let the radius of the second circle be \( R_2 \).
- Let the linear speed of the first car be \( v_a \).
- Let the linear speed of the second car be \( v_b \).
- Let the angular speed of the first car be \( \omega_a \).
- Let the angular speed of the second car be \( \omega_b \).
- Let the time taken to complete one revolution for both cars be \( T \).

2. **Relate Time, Speed, and Distance**:
- The distance traveled in one complete revolution for the first car is \( 2\pi R_1 \).
- The distance traveled in one complete revolution for the second car is \( 2\pi R_2 \).
- Therefore, the time taken for the first car is given by:
\[ T = \frac{2\pi R_1}{v_a} \]
- For the second car, the time taken is:
\[ T = \frac{2\pi R_2}{v_b} \]

3. **Set the Times Equal**:
- Since both cars take the same time to complete their circles, we can equate the two expressions for \( T \):
\[ \frac{2\pi R_1}{v_a} = \frac{2\pi R_2}{v_b} \]
- Cancel \( 2\pi \) from both sides:
\[ \frac{R_1}{v_a} = \frac{R_2}{v_b} \]

4. **Rearranging for Linear Speeds**:
- Rearranging the above equation gives:
\[ \frac{v_a}{v_b} = \frac{R_1}{R_2} \]
- This shows the ratio of linear speeds:
\[ \frac{v_a}{v_b} = \frac{R_1}{R_2} \]

5. **Relate Linear Speed and Angular Speed**:
- The relationship between linear speed and angular speed is given by:
\[ v_a = \omega_a R_1 \quad \text{and} \quad v_b = \omega_b R_2 \]

6. **Substituting in Ratios**:
- Substituting these into the ratio of linear speeds:
\[ \frac{\omega_a R_1}{\omega_b R_2} = \frac{R_1}{R_2} \]
- Simplifying gives:
\[ \frac{\omega_a}{\omega_b} = 1 \]

7. **Final Ratios**:
- Thus, the ratio of angular speeds is:
\[ \frac{\omega_a}{\omega_b} = 1 \]
- And the ratio of linear speeds is:
\[ \frac{v_a}{v_b} = \frac{R_1}{R_2} \]

### Summary of Results:
- The ratio of angular speeds \( \frac{\omega_a}{\omega_b} = 1 \)
- The ratio of linear speeds \( \frac{v_a}{v_b} = \frac{R_1}{R_2} \)