Boat and Stream Speed Calculation

This problem involves calculating the speed of the stream given the speed of the boat in still water and the distances traveled upstream and downstream in the same amount of time. Understanding the concepts of downstream and upstream motion is key to solving this.

Understanding Downstream and Upstream Motion

• Downstream: When the boat travels in the same direction as the stream, the speed of the stream adds to the speed of the boat. The effective speed is the sum of the boat's speed in still water and the stream's speed.
• Upstream: When the boat travels against the direction of the stream, the speed of the stream reduces the speed of the boat. The effective speed is the difference between the boat's speed in still water and the stream's speed.

Setting Up the Problem

Let's define the variables:

• Speed of the boat in still water (\(v_b\)) = 5 km/h (Given)
• Speed of the stream (\(v_s\)) = ? km/h (To be found)

Using these variables, we can express the downstream and upstream speeds:

• Speed downstream (\(v_d\)) = \(v_b + v_s = 5 + v_s\) km/h
• Speed upstream (\(v_u\)) = \(v_b - v_s = 5 - v_s\) km/h

We are given that the boat travels 26 km downstream and 14 km upstream in the same time.

Using the Time Equality Condition

The relationship between time, distance, and speed is:
Time = Distance / Speed

Let \(t_d\) be the time taken to travel downstream and \(t_u\) be the time taken to travel upstream.

• Time downstream (\(t_d\)) = Distance downstream / Speed downstream = \( \frac{26}{5 + v_s} \)
• Time upstream (\(t_u\)) = Distance upstream / Speed upstream = \( \frac{14}{5 - v_s} \)

Since the time taken for both journeys is the same (\(t_d = t_u\)), we can set up the equation:

\( \frac{26}{5 + v_s} = \frac{14}{5 - v_s} \)

Solving the Equation for Stream Speed

Now, we need to solve this equation for \(v_s\). We can do this by cross-multiplication:

\( 26 \times (5 - v_s) = 14 \times (5 + v_s) \)

Distribute the numbers on both sides:

\( (26 \times 5) - (26 \times v_s) = (14 \times 5) + (14 \times v_s) \)

\( 130 - 26v_s = 70 + 14v_s \)

Now, rearrange the terms to group \(v_s\) terms on one side and constant terms on the other:

\( 130 - 70 = 14v_s + 26v_s \)

\( 60 = 40v_s \)

Finally, solve for \(v_s\):

\( v_s = \frac{60}{40} \)

\( v_s = \frac{6}{4} \)

\( v_s = 1.5 \) km/h

The speed of the stream is 1.5 km/h.

Verification

Let's check if this stream speed works with the given information.

• If \(v_s = 1.5\) km/h and \(v_b = 5\) km/h:
• Speed downstream = \(5 + 1.5 = 6.5\) km/h
• Speed upstream = \(5 - 1.5 = 3.5\) km/h
• Time downstream = Distance / Speed = \(26 \text{ km} / 6.5 \text{ km/h} = 4\) hours
• Time upstream = Distance / Speed = \(14 \text{ km} / 3.5 \text{ km/h} = 4\) hours

Since the time taken is the same (4 hours), our calculated stream speed of 1.5 km/h is correct.

Revision Table: Key Formulas in Boat and Stream Problems

Additional Information on Boat and Stream Problems

Boat and stream problems are a common application of relative speed. The core idea is how the speed of the moving medium (stream/current) affects the speed of the object (boat/person) moving within it.

• Always assume the boat's speed in still water is greater than the stream's speed when traveling upstream. If the stream speed were greater, the boat would be carried backward.
• The time taken is often the connecting factor in these problems, allowing you to set up equations based on \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).
• These problems can be solved using the formulas directly if downstream and upstream speeds are given, or by setting up equations based on distances and times as demonstrated in this solution.
• Be careful with units (e.g., km/h, m/s). Ensure consistency throughout the calculation.

This type of problem is frequently asked in competitive exams and aptitude tests to check understanding of relative motion and basic algebraic equation solving.