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
A boat goes 30 km upstream in 3 hours and downstream in 1 hour. How much time (in hours) will this boat take to cover 60 km in still water?
The speed of a motorboat in still water is 20 km/h. It travels 150 km downstream and then returns to the starting point. If the round trip takes a total of 16 hours, what is the speed (in km/h) of the flow of river?
The time taken by a boat to travel 13 km downstream is the same as time taken by it to travel 7 km upstream. If the speed of the stream is 3 km/h, then how much time (in hours) will it take to travel a distance of 44.8 km in still water?
A man can row a distance of 8 km downstream in a certain time and can row 6 km upstream in the same time. If he rows 24 km upstream and the same distance downstream in \(1\frac{3}{4}\) hours, then the speed (in km/h) of the current is:
A boat goes 27 km upstream and 33 km downstream in 6 hours. In the same time it can go 36 km upstream and 22 km downstream. How much time will it take to go 36 km upstream and 44 km downstream?
