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 these

Answer

To solve the problem of finding the ratio of linear speeds of two buses A and B moving around 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 concentric circular paths with radii \( r_A \) and \( r_B \) respectively.
- Both buses complete their circular paths in the same time \( t \).

2. **Define Linear Speed:**
- The linear speed \( v \) of an object moving in a circular path is given by the formula:
\[ v = \frac{\text{Distance}}{\text{Time}} \]
- For a complete circular path, the distance traveled is the circumference of the circle.

3. **Calculate the Circumference:**
- The circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
- Therefore, the circumferences for buses A and B are:
- For bus A: \( C_A = 2\pi r_A \)
- For bus B: \( C_B = 2\pi r_B \)

4. **Calculate Linear Speeds:**
- The linear speed of bus A \( v_A \) is:
\[ v_A = \frac{C_A}{t} = \frac{2\pi r_A}{t} \]
- The linear speed of bus B \( v_B \) is:
\[ v_B = \frac{C_B}{t} = \frac{2\pi r_B}{t} \]

5. **Find the Ratio of Linear Speeds:**
- To find the ratio of the linear speeds \( \frac{v_A}{v_B} \):
\[ \frac{v_A}{v_B} = \frac{\frac{2\pi r_A}{t}}{\frac{2\pi r_B}{t}} = \frac{r_A}{r_B} \]
- The \( 2\pi \) and \( t \) cancel out, leaving us with the ratio of the radii.

6. **Conclusion:**
- 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 \( \frac{r_A}{r_B} \).