Understanding Average Speed in River Flow Problems

This problem asks us to find the average speed of a boat traveling the same distance upstream and downstream in a steady-flowing river. The key here is that the distance covered in both directions is the same, but the speeds are different.

Given Information:

• Upstream speed (\(v_{upstream}\)): 12 km/h
• Downstream speed (\(v_{downstream}\)): 24 km/h
• Distance traveled upstream is equal to the distance traveled downstream.

Calculating Average Speed for Equal Distances

When an object travels the same distance at two different speeds, the average speed is not simply the arithmetic mean of the two speeds. Instead, we use a specific formula derived from the definition of average speed (Total Distance / Total Time).

Let the distance traveled in one direction be \(d\). The total distance for the round trip is \(d + d = 2d\).

The time taken for the upstream journey (\(t_{upstream}\)) is:

\(t_{upstream} = \frac{\text{Distance}}{\text{Speed}} = \frac{d}{v_{upstream}} = \frac{d}{12}\)

The time taken for the downstream journey (\(t_{downstream}\)) is:

\(t_{downstream} = \frac{\text{Distance}}{\text{Speed}} = \frac{d}{v_{downstream}} = \frac{d}{24}\)

The total time (\(T\)) for the whole journey is:

\(T = t_{upstream} + t_{downstream} = \frac{d}{12} + \frac{d}{24}\)

To add these fractions, find a common denominator (24):

\(T = \frac{2d}{24} + \frac{d}{24} = \frac{2d + d}{24} = \frac{3d}{24} = \frac{d}{8}\)

The average speed (\(v_{average}\)) is Total Distance divided by Total Time:

\(v_{average} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{8}}\)

Simplify the expression:

\(v_{average} = 2d \times \frac{8}{d}\)

\(v_{average} = 2 \times 8\)

\(v_{average} = 16 \text{ km/h}\)

Alternatively, when the distance is the same for two parts of a journey with speeds \(v_1\) and \(v_2\), the average speed can be directly calculated using the formula:

\(v_{average} = \frac{2 v_1 v_2}{v_1 + v_2}\)

Substitute the given speeds \(v_1 = 12 \text{ km/h}\) and \(v_2 = 24 \text{ km/h}\):

\(v_{average} = \frac{2 \times 12 \times 24}{12 + 24}\)

\(v_{average} = \frac{2 \times 288}{36}\)

\(v_{average} = \frac{576}{36}\)

\(v_{average} = 16 \text{ km/h}\)

Both methods yield the same result.

Summary of Average Speed Calculation

Revision Table: Average Speed Concepts

Additional Information: Boat Speed and River Flow

In problems involving boats and rivers, we often consider the boat's speed in still water and the speed of the river current.

• Let \(v_b\) be the speed of the boat in still water.
• Let \(v_s\) be the speed of the stream (river current).

When the boat travels downstream, the current helps it, so the effective speed is:

\(v_{downstream} = v_b + v_s\)

When the boat travels upstream, the current opposes it, so the effective speed is:

\(v_{upstream} = v_b - v_s\)

In this problem, we were directly given the effective upstream and downstream speeds (12 km/h and 24 km/h). Had we been given \(v_b\) and \(v_s\), we would first calculate \(v_{upstream}\) and \(v_{downstream}\) and then use these values in the average speed formula for equal distances.

For example, from the given speeds:

• \(v_b - v_s = 12\)
• \(v_b + v_s = 24\)

Adding these two equations gives:

\((v_b - v_s) + (v_b + v_s) = 12 + 24\)

\(2v_b = 36\)

\(v_b = 18 \text{ km/h}\)

Substituting \(v_b = 18\) into \(v_b + v_s = 24\) gives:

\(18 + v_s = 24\)

\(v_s = 6 \text{ km/h}\)

So, the boat's speed in still water is 18 km/h, and the speed of the stream is 6 km/h. However, finding these speeds was not necessary to calculate the average speed for the total journey using the upstream and downstream speeds given.