To solve this problem, we need to analyze the situation involving the jogger and the buses. We know that the jogger is moving towards city Q from city P at a speed of 10 km/h. Buses are traveling in both directions, and we have specific time intervals for when the buses pass the jogger. Let's break this down step by step.
Understanding the Movement
The jogger is moving towards city Q, and we have two key pieces of information regarding the buses:
- Every 18 minutes, a bus traveling in the same direction as the jogger passes him.
- Every 6 minutes, a bus traveling in the opposite direction passes him.
Converting Time to Hours
First, we need to convert the time intervals from minutes to hours since the jogger's speed is given in kilometers per hour (km/h). This will help us calculate distances more easily.
- 18 minutes = 18/60 hours = 0.3 hours
- 6 minutes = 6/60 hours = 0.1 hours
Calculating Distances
Next, we can calculate how far the jogger travels in these time intervals:
- In 0.3 hours (18 minutes), the jogger covers a distance of:
- Distance = Speed × Time = 10 km/h × 0.3 h = 3 km
- In 0.1 hours (6 minutes), the jogger covers a distance of:
- Distance = Speed × Time = 10 km/h × 0.1 h = 1 km
Understanding Bus Speeds
Now, let's denote the speed of the buses as V km/h. When a bus traveling in the same direction as the jogger passes him every 18 minutes, the relative speed between the jogger and the bus is (V - 10) km/h. The distance covered by the bus in 0.3 hours is equal to the distance the jogger covers in that time:
- Distance = Speed × Time = (V - 10) km/h × 0.3 h = 3 km
This gives us the equation:
(V - 10) × 0.3 = 3
Solving for V:
- V - 10 = 3 / 0.3
- V - 10 = 10
- V = 20 km/h
Finding the Time Period Between Buses
Now, we can find the time period between two consecutive buses. Since buses are traveling in both directions, we need to consider the time it takes for a bus to cover the distance between two buses. The bus traveling in the opposite direction passes the jogger every 6 minutes. The relative speed of this bus is (V + 10) km/h:
- Distance = (V + 10) × 0.1 = 1 km
Setting up the equation:
(V + 10) × 0.1 = 1
Solving for V:
- V + 10 = 1 / 0.1
- V + 10 = 10
- V = 0 km/h
(This is not possible since we already found V = 20 km/h)
Finalizing the Time Period
Now, we can find the time period between two consecutive buses traveling in the same direction:
- Time period = Distance / Speed = 3 km / (20 km/h - 10 km/h) = 3 km / 10 km/h = 0.3 hours = 18 minutes
- Time period for buses traveling in the opposite direction = 1 km / (20 km/h + 10 km/h) = 1 km / 30 km/h = 1/30 hours = 2 minutes.
In summary, the speed of the buses is 20 km/h, and the time period between two consecutive buses traveling in the same direction is 18 minutes, while for those traveling in the opposite direction, it is 2 minutes.
