X and Y are two stations that are 280 km apart. A train starts at a certain time from X and travels towards Y at 60 km/h. After 2 hours, another train starts from Y and travels towards X at 20 km/h After how many hours does the train leaving from X meet the train which left from Y?
4 hours
Solving the Train Meeting Problem
This problem involves two trains traveling towards each other from different stations. We need to determine when they meet, considering that one train starts earlier than the other.
Understanding the Problem Setup
- Station X and Station Y are 280 km apart.
- Train A starts from X towards Y at a speed of 60 km/h.
- Train B starts from Y towards X at a speed of 20 km/h.
- Train A starts 2 hours earlier than Train B.
- We need to find the time elapsed from when Train X started until they meet.
Step-by-Step Solution
Step 1: Calculate the distance covered by Train A in the first 2 hours
Train A travels for 2 hours before Train B starts. The distance covered by Train A in this time is:
\( \text{Distance} = \text{Speed} \times \text{Time} \)
\( \text{Distance covered by Train A} = 60 \text{ km/h} \times 2 \text{ h} = 120 \text{ km} \)
Step 2: Calculate the remaining distance between the trains when Train B starts
The initial distance between X and Y was 280 km. After Train A has covered 120 km, the remaining distance between the two trains is:
\( \text{Remaining Distance} = \text{Total Distance} - \text{Distance covered by Train A} \)
\( \text{Remaining Distance} = 280 \text{ km} - 120 \text{ km} = 160 \text{ km} \)
Step 3: Calculate the relative speed of the two trains
Since the two trains are moving towards each other, their relative speed is the sum of their individual speeds.
\( \text{Relative Speed} = \text{Speed of Train A} + \text{Speed of Train B} \)
\( \text{Relative Speed} = 60 \text{ km/h} + 20 \text{ km/h} = 80 \text{ km/h} \)
This is the speed at which the distance between them is decreasing.
Step 4: Calculate the time taken for the trains to meet after Train B starts
The trains need to cover the remaining 160 km at a relative speed of 80 km/h. The time taken to meet after Train B starts is:
\( \text{Time to Meet (after Train B starts)} = \frac{\text{Remaining Distance}}{\text{Relative Speed}} \)
\( \text{Time to Meet} = \frac{160 \text{ km}}{80 \text{ km/h}} = 2 \text{ hours} \)
Step 5: Calculate the total time from when Train A started until they meet
Train A started 2 hours earlier than Train B. The trains meet 2 hours after Train B started. Therefore, the total time from when Train A started is:
\( \text{Total Time} = \text{Time Train A traveled before Train B starts} + \text{Time until they meet after Train B starts} \)
\( \text{Total Time} = 2 \text{ hours} + 2 \text{ hours} = 4 \text{ hours} \)
The train leaving from X meets the train which left from Y after a total of 4 hours from the time Train X departed.
| Event | Time Elapsed (from Train X start) | Distance Covered by Train X | Distance Covered by Train Y | Distance between Trains |
|---|---|---|---|---|
| Train X starts | 0 hours | 0 km | 0 km | 280 km |
| Train Y starts | 2 hours | \(60 \text{ km/h} \times 2 \text{ h} = 120 \text{ km}\) | 0 km | \(280 - 120 = 160 \text{ km}\) |
| Trains meet (after Train Y starts) | 2 more hours (Total 4 hours) | \(120 \text{ km} + 60 \text{ km/h} \times 2 \text{ h} = 120 + 120 = 240 \text{ km}\) | \(20 \text{ km/h} \times 2 \text{ h} = 40 \text{ km}\) | \(280 - (240+40) = 0 \text{ km}\) |
At the meeting point, Train X has covered 240 km from X, and Train Y has covered 40 km from Y. The sum of distances is \(240 + 40 = 280\) km, which is the total distance between X and Y.
Revision Table: Key Concepts
| Concept | Description | Formula |
|---|---|---|
| Distance, Speed, Time Relation | Relationship between distance, speed, and time. | \( \text{Distance} = \text{Speed} \times \text{Time} \) |
| Relative Speed (Approaching) | When two objects move towards each other, their relative speed is the sum of their individual speeds. | \( \text{Relative Speed} = \text{Speed 1} + \text{Speed 2} \) |
Additional Information: Types of Time and Distance Problems
Problems involving time and distance often fall into different categories:
- Basic Problems: Directly applying the \( \text{Distance} = \text{Speed} \times \text{Time} \) formula.
- Relative Speed Problems: Involving objects moving towards or away from each other.
- Average Speed Problems: Calculating the average speed over a journey with varying speeds.
- Train Problems: Specific cases involving the length of trains, platforms, etc.
- Boat and Stream Problems: Considering the effect of water current on speed.
Understanding the concept of relative speed is crucial for solving problems where multiple objects are in motion simultaneously, like this train meeting problem.
Important Questions from Relative Speed
The driver of a car, which is travelling at a speed of 75 km/h, locates a bus 80 m ahead of him, travelling in the same direction. After 15 seconds, he finds that the bus is 40 m behind the car. What is the speed of the bus (in km/h)?
The distance between two station A and B is 800 km. A train X starts from A and moves towards B at 40 km/h and another trains Y starts from B and moves towards A at 60 km/h. how far from A will they cross each other?
A and B are travelling towards each other from the points P and Q respectively. After crossing each other, A and B take \(6\frac{1}{8}\) hours and 8 hours, respectively, to reach their destinations Q and P, respectively. If the speed of B is 16.8 km/h, then the speed (in km/hr) of A is:
A train leaves station A at 8 am and reaches station B at 12 noon. A car leaves station B at 8:30 am and reaches station A at the same time when the train reaches station B. At what time do they meet?
A and B start moving towards each other from places X and Y respectively, at the same time. The speed of A is 20% more than that of B. After meeting on the way, A and B take \(2\frac{1}{2}\) hours and x hours now to reach Y and X respectively. What is the value of x?
