Two buses start from opposite points towards each other on a main road, 185 km apart. The first bus runs for 35 km and takes a right turn and runs for 17 km. It then turns left and runs for another 42 km and takes the direction back to reach the main road, in the meantime, due to a minor breakdown, the other bus has run only 36 km along the main road. What would the distance between the two buses be at this point?
72 km
Solving the Bus Distance Problem
This problem involves two buses moving towards each other from opposite points on a main road. We need to determine the distance between them after they have traveled for a certain amount of time, considering one bus takes a detour.
Understanding the Initial Setup
The main road connects two points that are 185 km apart. Let's call the starting point of the first bus A and the starting point of the second bus B. The total distance between A and B is 185 km.
Analyzing the First Bus's Journey
The first bus starts from point A and travels as follows:
- Runs for 35 km along the main road towards B.
- Takes a right turn and runs for 17 km (perpendicular to the main road).
- Turns left and runs for another 42 km (parallel to the main road).
- Takes the direction back to reach the main road. This means it turns right and travels 17 km perpendicular to the main road again, returning to the main road.
To find out how far the first bus is from its starting point A along the main road, we need to consider its displacement along the main road direction. The segments perpendicular to the main road (17 km out and 17 km back) cancel each other out in terms of displacement along the main road.
The total distance covered by the first bus along the direction parallel to the main road is the initial 35 km plus the 42 km segment. So, the first bus ends up on the main road at a point that is $35 \text{ km} + 42 \text{ km} = 77 \text{ km}$ from its starting point A.
Its position is 77 km from point A on the main road.
Analyzing the Second Bus's Journey
The second bus starts from point B (185 km from A) and travels towards A along the main road.
- Due to a minor breakdown, the second bus travels only 36 km along the main road towards A.
Its position is 36 km from point B on the main road.
Calculating the Distance Between the Buses
The initial distance between the two buses (points A and B) is 185 km.
The first bus is now 77 km from A.
The second bus is now 36 km from B.
Since they started from opposite points and moved towards each other, the distance between them at this point is the total initial distance minus the distance covered by each bus towards the other.
Distance between buses = Total distance AB - (Distance covered by Bus 1 from A) - (Distance covered by Bus 2 from B)
Using the values we found:
Distance = $185 \text{ km} - 77 \text{ km} - 36 \text{ km}$
Distance = $108 \text{ km} - 36 \text{ km}$
Distance = $72 \text{ km}$
The distance between the two buses at this point is 72 km.
| Detail | Distance (km) |
|---|---|
| Total initial distance between A and B | 185 |
| Bus 1's effective distance from A along main road | $35 + 42 = 77$ |
| Bus 2's distance from B along main road | 36 |
| Distance between buses | $185 - 77 - 36 = 72$ |
Conclusion
By analyzing the path of each bus and calculating their final positions relative to their starting points on the main road, we found that the distance separating them is 72 km.
Revision Table: Key Concepts Review
Reviewing the steps and concepts used in this distance calculation problem:
- Displacement vs. Distance: Bus 1 traveled a longer distance (35+17+42+17 = 111 km), but its displacement along the main road is only 77 km. We used displacement along the main road to find its effective position relative to its starting point for calculating the distance between the two buses.
- Relative Motion: When objects move towards each other, the distance between them decreases by the sum of the distances each object covers.
- Calculating Remaining Distance: The distance between two points is the total distance minus the segments covered from each end towards the middle.
Additional Information: Related Distance Problems
Similar problems often involve:
- Calculating average speed or time taken.
- Meeting points of objects moving towards or away from each other.
- Problems involving varying speeds or multiple segments of travel.
- Understanding vector displacement versus scalar distance.
These types of problems are common in quantitative aptitude and physics, requiring careful attention to the direction and distance covered by each object.
Important Questions from Speed Time and Distance
A journey of 900 km is completed in 11 h. If two-fifth of the journey is completed at the speed of 60 km/h, at what speed (in km/h) is the remaining journey completed?
A car starts from point A towards point B, travelling at the speed of 20 km/h. 1 \(\frac{1}{2}\) hours later, another car starts from point A and travelling at the speed of 30 km/h and reaches 2 \(\frac{1}{2}\) hours before the first car. Find the distance between A and B.
A bus covered a distance of 162 km. If speed of this bus is 15 m/s, then what will be the time taken ?
An athlete runs an 800 m race in 96 seconds. His speed (in km / h) is:
A person has to cover a distance of 150 km in 15 hours. If he traveled with the speed of 11.8 km/hr for 10 hours. At what speed he has to travel to cover the remaining distance in the remaining time?
