Two buses A and B are moving around concentric circular pathe of radii `r_(A)` and `r_(B)` If the two buses complete the circular paths in the sme time. The ratio on their linear speeds is

Options

A. 1
B. `r_A //r_B`
C. `r_B//r_A`
D. none of the above

Answer

To solve the problem of finding the ratio of the linear speeds of two buses A and B moving along concentric circular paths of radii \( r_A \) and \( r_B \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have two buses, A and B, moving in circular paths with radii \( r_A \) and \( r_B \) respectively.
- Both buses complete their circular paths in the same amount of time.

2. **Define Linear Speed**:
- The linear speed \( v \) of an object moving in a circular path is given by the formula:
\[ v = \frac{2\pi r}{T} \]
where \( r \) is the radius of the circular path and \( T \) is the time period (the time taken to complete one full revolution).

3. **Set Up the Equations**:
- For bus A, the linear speed \( v_A \) can be expressed as:
\[ v_A = \frac{2\pi r_A}{T} \]
- For bus B, the linear speed \( v_B \) can be expressed as:
\[ v_B = \frac{2\pi r_B}{T} \]

4. **Find the Ratio of Linear Speeds**:
- To find the ratio of the linear speeds \( \frac{v_A}{v_B} \), we can divide the two equations:
\[ \frac{v_A}{v_B} = \frac{\frac{2\pi r_A}{T}}{\frac{2\pi r_B}{T}} \]
- The \( 2\pi \) and \( T \) cancel out:
\[ \frac{v_A}{v_B} = \frac{r_A}{r_B} \]

5. **Conclusion**:
- Therefore, the ratio of the linear speeds of buses A and B is:
\[ \frac{v_A}{v_B} = \frac{r_A}{r_B} \]

### Final Answer:
The ratio of their linear speeds is \( \frac{r_A}{r_B} \).