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.

Revision Table: Relative Speed Concepts

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.