Two cities A and B are connected by a regular bus servicewith a bus leaving in either directionin every T min. a man cycling with a speed of 20 km/ph from A to B notices that a bus past him in every 18 minutes in the direction of his motion and in every 6 minutes in opposite direction.What is the period T of the service and with what speed (assumed constants) do the buses ply on the road?

To solve this problem, we need to analyze the situation involving the cyclist and the buses traveling between cities A and B. Let's break it down step by step to find the period T of the bus service and the speed of the buses.

Understanding the Scenario

We have a cyclist moving from A to B at a speed of 20 km/h. Buses leave both cities at regular intervals of T minutes. The cyclist observes that a bus passes him every 18 minutes in the direction he is cycling and every 6 minutes when coming from the opposite direction.

Setting Up the Equations

Let’s denote the speed of the buses as V km/h. We need to convert the time intervals into hours for consistency in units:

• 18 minutes = 18/60 hours = 0.3 hours
• 6 minutes = 6/60 hours = 0.1 hours

Analyzing the Buses Coming from Behind

When a bus is traveling in the same direction as the cyclist, the relative speed between the bus and the cyclist is (V - 20) km/h. The time taken for the bus to catch up with the cyclist is 0.3 hours. Using the formula for distance (Distance = Speed × Time), we can express this as:

Distance = (V - 20) × 0.3

Analyzing the Buses Coming from the Opposite Direction

For the buses coming from the opposite direction, the relative speed is (V + 20) km/h. The time taken for the bus to reach the cyclist is 0.1 hours. Thus, we can write:

Distance = (V + 20) × 0.1

Equating the Distances

Since both distances represent the same distance between two consecutive buses, we can set the two equations equal to each other:

(V - 20) × 0.3 = (V + 20) × 0.1

Solving for V

Now, let’s solve this equation:

• 0.3V - 6 = 0.1V + 2
• 0.3V - 0.1V = 2 + 6
• 0.2V = 8
• V = 40 km/h

Finding the Period T

Now that we have the speed of the buses (V = 40 km/h), we can find the period T. The time between buses is determined by the distance between them divided by the speed of the buses. The distance between two buses can be calculated as:

Distance = Speed × Time = 40 × T/60

Using the distance formula for the buses passing the cyclist:

Distance = (V - 20) × 0.3 = (40 - 20) × 0.3 = 6 km

Setting the two expressions for distance equal gives us:

40 × T/60 = 6

Now, solving for T:

• 40T = 6 × 60
• 40T = 360
• T = 9 minutes

Final Results

In summary, the period T of the bus service is 9 minutes, and the speed of the buses is 40 km/h. This analysis shows how relative motion can be used to solve problems involving different speeds and time intervals.