Finding the Speed of a Water Current Using Boat Speed and Distance

The problem of determining the speed of a water current is a classic application in physics and mathematics. By leveraging the given information and applying basic principles, we can accurately determine the speed of the water current. In this article, we will walk through a detailed example to solve such problems. We will introduce the concepts, provide a step-by-step solution, and identify the key elements that are crucial for understanding this type of problem.

Understanding the Problem

Given a boat with a certain speed in still water and a known ratio of its speed to the water current, we need to determine the speed of the water current. Additionally, the boat's travel distance and time are provided. The steps to solve this problem are detailed as follows:

Step-by-Step Solution

Consider the following information:

Ratio of the speed of the boat downstream to the speed of the water current: 5:1 The boat travels 12.6 km upstream in 84 minutes

We will define the variables:

n- bn- c: Speed of the boat in still water in km/h n- c: Speed of the water current in km/h

Setting Up the Equations

The first piece of information given is the ratio of the speed of the boat downstream to the speed of the water current. We can express this as:

n- Downstream speed b c 5c

From this, we can express the speed of the boat in still water in terms of the speed of the water current:

b 5c – c 4c

Upstream Speed Calculation

Next, we know that the boat travels 12.6 km upstream in 84 minutes. First, we convert 84 minutes to hours:

84 minutes 84/60 hours 1.4 hours

The upstream speed is given by:

n- Upstream speed b – c

The upstream speed can be calculated using the distance and time:

n- Upstream speed 12.6 km / 1.4 hours 9 km/h

Solving the Equations

We have two equations:

b – c 9 b 4c

Substituting the second equation into the first:

4c – c 9

3c 9

c 3 km/h

Thus, the speed of the water current is 3 km/h.

Summary

The speed of the water current c is 3 km/h. This solution involves using the given ratio of speeds, converting time to a consistent unit, and solving the system of equations to find the unknown variables.

Additional Examples

Let's consider an additional example to solidify our understanding:

Statement of the problem: The respective ratio of speed of a boat and that of the water current is 5:1. If the boat can travel 12.6 km upstream in 84 minutes, what is the speed of the water current?

From the given information:

Let the speed of the boat in still water be x and the speed of the current be y? Using the ratio and the distance and time relationship:

We can express the speed of the boat in terms of the water current as:

x 4y

The upstream speed is:

x - y 3y 9 km/h

Hence, the speed of the water current y is:

y 3 km/h

The speed of the boat in still water x is:

x 12 km/h

Conclusion

Understanding and solving such problems effectively requires a clear understanding of the relationships between variables and the application of basic algebra. The steps involved are:

Identify the given ratios and variables. Set up the equations based on the given information. Solve the equations to find the unknowns.

By practicing similar problems, one can master the techniques and improve their problem-solving skills in physics and mathematics.