Two cities `A and B` are connected by a regular bus service with buses plying in either direction every `T` seconds. The speed of each bus is uniform and equal to `V_b`. A cyclist cycles from `A to B` with a uniform speed of `V_c`. A bus goes past the cyclist in `T_1` second in the direction `A to B` and every `T_2` second in the direction `B to A`. Then

Options

A. `T_1 = (V_b T)/(V_b + V_c)`
B. `T_2 = (V_b T)/(V_b - V_c)`
C. `T_1 = (V_b T)/(V_b - V_c)`
D. `T_2 = (V_b T)/(V_b + V_c)`

Explanation

To solve the problem, we will analyze the situation step by step, focusing on the relative motion between the cyclist and the buses traveling between cities A and B.

### Step 1: Understanding the Problem
- We have two cities, A and B, connected by a bus service.
- Buses travel from A to B and from B to A every T seconds.
- The speed of the buses is \( V_b \).
- A cyclist travels from A to B at a speed of \( V_c \).
- We need to find the time taken for a bus to pass the cyclist in both directions.

### Step 2: Distance Between Buses
- The distance between two consecutive buses traveling in the same direction (either A to B or B to A) can be calculated as:
\[ \text{Distance} = V_b \times T \]
- This distance is the same for both directions since the buses are spaced evenly.

### Step 3: Analyzing the Bus Passing the Cyclist from A to B
- When a bus passes the cyclist traveling from A to B, the relative speed of the bus with respect to the cyclist is:
\[ V_{\text{relative}} = V_b - V_c \]
- The time taken for the bus to pass the cyclist (denoted as \( T_1 \)) can be expressed as:
\[ T_1 = \frac{\text{Distance between buses}}{\text{Relative speed}} = \frac{V_b \times T}{V_b - V_c} \]

### Step 4: Analyzing the Bus Passing the Cyclist from B to A
- When a bus is traveling from B to A, the relative speed of the bus with respect to the cyclist is:
\[ V_{\text{relative}} = V_b + V_c \]
- The time taken for the bus to pass the cyclist in this case (denoted as \( T_2 \)) can be expressed as:
\[ T_2 = \frac{\text{Distance between buses}}{\text{Relative speed}} = \frac{V_b \times T}{V_b + V_c} \]

### Step 5: Conclusion
- We have derived the expressions for \( T_1 \) and \( T_2 \):
- \( T_1 = \frac{V_b T}{V_b - V_c} \)
- \( T_2 = \frac{V_b T}{V_b + V_c} \)

### Final Answer
- The correct expressions for the times are:
- \( T_1 = \frac{V_b T}{V_b - V_c} \) (Option C)
- \( T_2 = \frac{V_b T}{V_b + V_c} \) (Option D)