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Two buses A and B are moving around conc...

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

A

1

B

`r_A/r_B`

C

`r_B/r_A`

D

None of these

Text Solution

AI Generated Solution

The correct Answer is:
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 \( \frac{r_A}{r_B} \).
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