Determining the Speed of a Boat in Still Water: A Real-World Puzzle

Boat speed is a fascinating topic in physics and mathematics, especially when considering the effects of the water's current. This guide delves into a particular puzzle faced by boat operators regarding the boat's speed in still water under specific conditions.

The Problem

A boat takes twice as long to travel upstream as it does to travel downstream when the speed of water is 3 kilometers per hour. The challenge is to determine the speed of the boat in still water. We begin by setting up the problem with appropriate variables and equations.

Solving the Problem

Let’s denote the speed of the boat in still water as ( b ) kilometers per hour and the speed of the water as ( w 3 ) kilometers per hour. When the boat is going downstream, its effective speed is ( b w b 3 ) kilometers per hour. Similarly, when the boat is going upstream, its effective speed is ( b - w b - 3 ) kilometers per hour.

According to the problem, the time taken to travel downstream is double the time taken to travel upstream. Given that the distances traveled both upstream and downstream are the same, we can use the formula for time, which is distance divided by speed:

Time taken to travel downstream ( t_d frac{d}{b 3} )

Time taken to travel upstream ( t_u frac{d}{b - 3} )

Given that the time taken upstream is twice the time taken downstream, we have:

( t_d 2 times t_u )

Substituting the expressions for ( t_d ) and ( t_u ), we get:

( frac{d}{b 3} 2 times frac{d}{b - 3} )

Since the distance ( d ) is the same on both sides, we can cancel it out (assuming ( d eq 0 )), leading to:

( frac{1}{b 3} frac{2}{b - 3} )

Cross-multiplying the terms, we obtain:

( b - 3 2(b 3) )

Expanding and simplifying the equation, we have:

( b - 3 2b 6 )

( b - 2b 6 3 )

( -b 9 )

( b 9 )

The speed of the boat in still water cannot be negative, hence the correct solution is ( b 9 ) kilometers per hour.

Verification

To verify our solution, let’s plug ( b 9 ) into the speeds:

Downstream speed: ( 9 3 12 ) km/h

Upstream speed: ( 9 - 3 6 ) km/h

It is clear that 12 km/h is indeed twice 6 km/h, confirming our solution.

Conclusion

The problem demonstrates the importance of algebraic reasoning and the inverse relationship between speed and time when traveling upstream and downstream. The calculated speed of the boat in still water is 9 km/h, offering a practical application of physics and mathematics.

Understanding such concepts can help in effective navigation and planning, ensuring that boats travel efficiently and safely in various water conditions.