The Speed of the River Current: A Problem in Ship Navigation

When navigating a river, the speed of the river current plays a crucial role in determining the overall velocity of a boat. This article explores a specific problem related to a boat crossing a river and introduces a detailed step-by-step solution to determine the speed of the river current. Understanding these concepts can help improve navigation strategies and enhance the safety and efficiency of vessel operations.

Problem Statement

A boat travels with a speed of 5 km/h in still water and crosses a river of width 1 km along the shortest possible path in 15 minutes. The goal is to calculate the velocity of the river water in km/h. Let's break down the problem and solve it step-by-step.

Step-by-Step Solution

Step 1: Determine the Speed Across the River

The boat crosses the river in the shortest possible path, meaning it aims directly across it, while being carried downstream by the river current. Let's denote the speed of the boat in still water as v_b 5 km/h, the width of the river as d 1 km, and the time taken to cross the river as t 15 minutes.

First, convert the time from minutes to hours:

0.25 hours  15 minutes / 60

Calculate the speed across the river, denoted as v_across:

v_across  1 km / 0.25 hours  4 km/h

Step 2: Set Up the Relationship Between the Boat's Speed and the River's Current

Let V_r be the velocity of the river current in km/h. The boat's actual velocity has two components:

The velocity across the river, v_across 4 km/h, perpendicular to the current. The velocity downstream due to the river's current, V_r.

Using the Pythagorean theorem, where the boat's speed in still water is the hypotenuse:

5^2  4^2   V_r^2

Solve for V_r:

25  16   V_r^2V_r^2  9V_r  sqrt{9}  3 km/h

Conclusion

The velocity of the river water is 3 km/h.

Alternative Solution Approach

As an additional method, let's consider the boat moving at an angle X with the width of the river, facing a little in the direction of the river's flow. In this setup:

The boat's velocity perpendicular to the river, V_{boat} cos X 5 cos X, enables it to cross the river directly to the opposite bank. The velocity parallel to the river, V_{boat} sin X 5 sin X, is opposed by the river's current, V_w.

Given the time taken to cross the river, 15/60 0.25 hours, we have:

cos X  4/5sin X  3/5V_w  5 sin X  5 * (3/5)  3 km/h

Summary and Importance

This problem showcases the practical application of vector addition and trigonometry in understanding the dynamics of ship navigation. By accurately determining the river's current speed, navigators can make more informed decisions, ensuring safe and efficient travel. Understanding these concepts is crucial for both recreational and professional maritime operations.

Conclusion

The velocity of the river water is 3 km/h. This solution demonstrates the importance of considering the river's current in boat navigation and how mathematical principles can be applied to real-world problems.