Two boats go downstream in a river at a speed of 35 km/hr and 25 km/hr. The speed of the stream is 4 km/hr. The distance between both boats initially is 60 km. In how much time will they meet if the boats are going downstream?
6 hours
Understanding the Problem: Boats Meeting Downstream
The question asks for the time it takes for two boats, moving downstream in a river, to meet, given their downstream speeds and initial distance.
Here's what we know:
- Speed of Boat 1 downstream: 35 km/hr
- Speed of Boat 2 downstream: 25 km/hr
- Initial distance between the boats: 60 km
- The boats are both moving in the same direction (downstream).
Note: The speed of the stream (4 km/hr) is provided, but it's not needed to calculate the relative speed between the boats since their speeds relative to the river bank (downstream speeds) are already given.
Applying the Concept of Relative Speed
When two objects move in the same direction, their relative speed is the difference between their individual speeds. This relative speed tells us how quickly the distance between them is changing.
In this scenario, the faster boat (Boat 1) is trying to catch up to the slower boat (Boat 2). The rate at which the faster boat closes the distance is their relative speed.
Calculating the Relative Speed
The relative speed of Boat 1 with respect to Boat 2 is the difference between their downstream speeds:
\(\text{Relative Speed} = \text{Speed of Boat 1 downstream} - \text{Speed of Boat 2 downstream}\)
\(\text{Relative Speed} = 35 \text{ km/hr} - 25 \text{ km/hr}\)
\(\text{Relative Speed} = 10 \text{ km/hr}\)
This means the distance between the boats decreases by 10 km every hour.
Calculating the Time to Meet
The time it takes for the faster boat to cover the initial distance between them, using their relative speed, will be the time until they meet. The formula is:
\(\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}}\)
We know the initial distance is 60 km and the relative speed is 10 km/hr.
\(\text{Time} = \frac{60 \text{ km}}{10 \text{ km/hr}}\)
\(\text{Time} = 6 \text{ hours}\)
Conclusion
The boats will meet in 6 hours.
| Parameter | Value |
|---|---|
| Speed of Boat 1 (Downstream) | 35 km/hr |
| Speed of Boat 2 (Downstream) | 25 km/hr |
| Initial Distance | 60 km |
| Relative Speed (Same Direction) | 10 km/hr |
| Time to Meet | 6 hours |
Revision Table: Relative Speed Concepts
| Scenario | Relative Speed Calculation | Explanation |
|---|---|---|
| Objects moving in the same direction | Difference of speeds (\(|v_1 - v_2|\)) | The distance between them changes at a rate equal to the difference in their speeds. Useful for catch-up problems. |
| Objects moving in opposite directions | Sum of speeds (\(v_1 + v_2\)) | The distance between them decreases or increases at a rate equal to the sum of their speeds. Useful for meeting problems when starting from a distance apart and moving towards each other. |
Additional Information: Boats and Streams
In problems involving boats and streams, speeds are usually given relative to the water or relative to the ground (river bank). Key terms:
- Speed of Boat in Still Water: The speed the boat can travel without the effect of the current. Let's call this \(V_b\).
- Speed of Stream (or Current): The speed of the water flow. Let's call this \(V_s\).
- Downstream Speed: Speed of the boat when moving with the current. This is the speed relative to the river bank. \(V_{\text{downstream}} = V_b + V_s\).
- Upstream Speed: Speed of the boat when moving against the current. This is also the speed relative to the river bank. \(V_{\text{upstream}} = V_b - V_s\).
In this specific problem, the speeds given (35 km/hr and 25 km/hr) are already the downstream speeds relative to the bank, meaning they already incorporate the effect of the stream. That's why the stream speed (4 km/hr) wasn't directly used in the relative speed calculation between the two boats moving downstream.
Important Questions from Boat and River
- If a boat travels with a speed of 12 km/hr in still water and the speed of stream is 3 km/hr, what would be the time taken (in hours) by boat to go 60 km downstream?
- A boat moves at a speed of 7 km/h in still water. The boat goes 40 km downstream and then 40 km upstream. It is observed that the total time taken for this journey is 560 minutes less than the time it would take if the speed of the stream were 2 km/h faster. What is the initial speed (in km/h) of the stream?
- A boat travels 120 km downstream in 6 hours and 120 km upstream in 10 hours. The speed of the current is 4 km/hr. What is the speed of the boat in still water?
- Ashok rows 43.5 kmph in still water. If the river is running at 14.5 kmph, it takes him 90 minutes to row to a place and back. How far is the place from the starting point? (In km)
- Aman can go downstream thrice as fast as he can go upstream between two specific points on a river. If the river flows at 8 kmph, what is the speed of the boat in still water (in kmph) ?
