Solving the Boat and Stream Problem

This problem involves understanding the concepts of relative speed in the context of boat and stream motion. When a man rows downstream, the speed of the current adds to his speed in still water. When he rows upstream, the speed of the current opposes his speed in still water.

Defining Variables

• Let the speed of the man in still water be \(v_m\) km/h.
• Let the speed of the current be \(v_c\) km/h.

Understanding Downstream and Upstream Speeds

• Speed downstream = Speed of man in still water + Speed of current \( = v_m + v_c\).
• Speed upstream = Speed of man in still water - Speed of current \( = v_m - v_c\).

Note that for upstream motion to be possible, the speed of the man in still water must be greater than the speed of the current (\(v_m > v_c\)).

Setting Up Equations from the Given Information

We are given two pieces of information that allow us to set up equations:

1. A man rows 8 km downstream and 6 km upstream in the same amount of time.
2. He rows 24 km upstream and the same distance (24 km) downstream in a total of \(1\frac{3}{4}\) hours.

From the first condition (Equal Time):

Time = Distance / Speed.

Time taken to row 8 km downstream is \(T_1 = \frac{8}{v_m + v_c}\).

Time taken to row 6 km upstream is \(T_2 = \frac{6}{v_m - v_c}\).

Since \(T_1 = T_2\), we have:

\(\frac{8}{v_m + v_c} = \frac{6}{v_m - v_c}\)

Cross-multiplying gives:

\(8(v_m - v_c) = 6(v_m + v_c)\)

\(8v_m - 8v_c = 6v_m + 6v_c\)

Rearranging the terms to one side:

\(8v_m - 6v_m = 6v_c + 8v_c\)

\(2v_m = 14v_c\)

Dividing by 2, we get a relationship between \(v_m\) and \(v_c\):

\(v_m = 7v_c\) --- (Equation 1)

This means the man's speed in still water is 7 times the speed of the current.

From the second condition (Total Time):

The total time taken to row 24 km upstream and 24 km downstream is \(1\frac{3}{4}\) hours, which is \(\frac{7}{4}\) hours.

Time taken to row 24 km upstream = \(\frac{24}{v_m - v_c}\).

Time taken to row 24 km downstream = \(\frac{24}{v_m + v_c}\).

The sum of these times is the total time:

\(\frac{24}{v_m - v_c} + \frac{24}{v_m + v_c} = \frac{7}{4}\) --- (Equation 2)

Solving the System of Equations

Now we substitute Equation 1 (\(v_m = 7v_c\)) into Equation 2.

First, find the upstream and downstream speeds in terms of \(v_c\):

• Speed upstream \( = v_m - v_c = 7v_c - v_c = 6v_c\).
• Speed downstream \( = v_m + v_c = 7v_c + v_c = 8v_c\).

Substitute these into Equation 2:

\(\frac{24}{6v_c} + \frac{24}{8v_c} = \frac{7}{4}\)

Simplify the fractions on the left side:

\(\frac{4}{v_c} + \frac{3}{v_c} = \frac{7}{4}\)

Combine the terms on the left side:

\(\frac{4 + 3}{v_c} = \frac{7}{4}\)

\(\frac{7}{v_c} = \frac{7}{4}\)

To find \(v_c\), we can cross-multiply or notice that the numerators are equal, so the denominators must also be equal (assuming \(v_c \neq 0\), which it must be for a current to exist):

\(7 \times 4 = 7 \times v_c\)

\(28 = 7v_c\)

Divide by 7:

\(v_c = \frac{28}{7}\)

\(v_c = 4\)

The speed of the current is 4 km/h.

Verification

If \(v_c = 4\) km/h, then \(v_m = 7 \times 4 = 28\) km/h.

• Upstream speed \( = v_m - v_c = 28 - 4 = 24\) km/h.
• Downstream speed \( = v_m + v_c = 28 + 4 = 32\) km/h.

Check the first condition: Time for 8 km downstream = \(8/32 = 1/4\) hour. Time for 6 km upstream = \(6/24 = 1/4\) hour. This matches.

Check the second condition: Time for 24 km upstream = \(24/24 = 1\) hour. Time for 24 km downstream = \(24/32 = 3/4\) hour. Total time = \(1 + 3/4 = 7/4 = 1\frac{3}{4}\) hours. This also matches.

Final Answer

The speed of the current is 4 km/h.

Revision Table: Boat and Stream Concepts

Additional Information: Relative Speed in Fluids

The problems involving boats and streams are practical applications of the concept of relative speed. When an object moves in a fluid (like water or air) that is also moving, its speed relative to a stationary point on the ground (or outside the fluid) is the vector sum of its speed relative to the fluid and the speed of the fluid itself.

• Moving with the fluid (Downstream/Tailwind): The speeds add up. Relative speed = Speed of object in still fluid + Speed of fluid.
• Moving against the fluid (Upstream/Headwind): The fluid's speed is subtracted from the object's speed in still fluid. Relative speed = Speed of object in still fluid - Speed of fluid.

This principle is used not only in boat and stream problems but also in aircraft navigation where wind speed affects the plane's ground speed.