Two towns A and B are connected by a regular bus service with a bus leaving in either direction every T minutes. A man cycling with a speed of 20 km h^-1 in the direction A to B notices that a bus goes past him every 18 min
in the direction of his motion, and every 6 min in the opposite direction. The period T of the bus service and speed of buses on the road are

In this problem, two towns, A and B, are connected by a regular bus service. The goal is to find the period T of the bus service and the speed V of the buses. A cyclist moving at a constant speed of 20 km/h observes buses passing him every 18 minutes in the same direction (A to B) and every 6 minutes in the opposite direction (B to A).

Step 1: Understand the Problem

The cyclist moves from town A to town B at a speed of 20 km/h. We need to determine the period T of the bus service and the speed V of the buses connecting the two towns A and B.

Step 2: Set Up Relative Speeds

• Speed of the cyclist: 20 km/h
• Speed of the buses: V km/h

When buses move in the same direction as the cyclist (from A to B):

• Relative speed of the bus with respect to the cyclist = V - 20 km/h.

When buses move in the opposite direction (from B to A):

• Relative speed of the bus with respect to the cyclist = V + 20 km/h.

Step 3: Derive the Distance Between Two Buses

The distance D between two consecutive buses can be expressed as:

• D = V × T

Using the observed time intervals:

• For buses traveling from A to B: D = (V - 20) × (18/60)
• For buses traveling from B to A: D = (V + 20) × (6/60)

Step 4: Equating Distances

Since the distances are equal for both directions:

(V - 20) × (18/60) = (V + 20) × (6/60)

Step 5: Solve the Equation

1. Eliminate the common factor (1/60):

   (V - 20) × 18 = (V + 20) × 6

2. Expand both sides:

   18V - 360 = 6V + 120

3. Rearrange to find V:

   18V - 6V = 360 + 120

   12V = 480

   V = 40 km/h

Step 6: Determine the Period T

Now that V is known, substitute into the distance equation to find T:

D = V × T

40 × T = (40 - 20) × (18/60)

Calculate:

D = 20 × 0.3 = 6 km

40 × T = 6

T = 6 / 40 = 0.15 hours = 9 minutes

Final Answer

• Speed of the buses connecting two towns A and B: 40 km/h
• Period of the bus service: 9 minutes

This solution highlights the periodic nature of the bus service and the speeds involved between two towns A and B.