Rate of the boat in still water

Oct 13, 2025 by talynt.org 107 views

A Motorboat's Journey: Uncovering the Rate of the Boat in Still Water and the Rate of the Current

In this article, we will delve into the world of mathematics and explore the concept of relative motion. We will use the scenario of a motorboat traveling upstream and downstream to determine the rate of the boat in still water and the rate of the current. This problem is a classic example of how to apply the concept of relative motion to real-world situations.

Let's break down the problem and understand what is being asked. We are given that a motorboat takes 3 hours to travel 108 kilometers going upstream. This means that the boat is traveling against the current, and its speed is reduced due to the opposing force of the current. On the other hand, the return trip takes 2 hours going downstream, which means the boat is traveling with the current, and its speed is increased due to the assisting force of the current.

Let's define the variables involved in this problem:

B: The rate of the boat in still water (in km/h)
C: The rate of the current (in km/h)
D: The distance traveled by the boat (in km)
T: The time taken by the boat to travel upstream (in hours)
T': The time taken by the boat to travel downstream (in hours)

We can now formulate the equations based on the given information:

Upstream: The boat's speed is reduced due to the opposing force of the current, so its effective speed is B - C. The distance traveled is 108 km, and the time taken is 3 hours, so we can write the equation: D / (B - C) = T
Downstream: The boat's speed is increased due to the assisting force of the current, so its effective speed is B + C. The distance traveled is the same (108 km), and the time taken is 2 hours, so we can write the equation: D / (B + C) = T'

We can simplify the equations by substituting the given values:

Upstream: 108 / (B - C) = 3
Downstream: 108 / (B + C) = 2

We can now solve the equations to find the values of B and C.

First, let's solve the upstream equation:

108 / (B - C) = 3

Multiplying both sides by (B - C), we get:

108 = 3(B - C)

Expanding the right-hand side, we get:

108 = 3B - 3C

Now, let's solve the downstream equation:

108 / (B + C) = 2

Multiplying both sides by (B + C), we get:

108 = 2(B + C)

Expanding the right-hand side, we get:

108 = 2B + 2C

We now have two equations with two variables:

108 = 3B - 3C 108 = 2B + 2C

We can solve these equations simultaneously to find the values of B and C.

First, let's multiply the first equation by 2 and the second equation by 3:

216 = 6B - 6C 324 = 6B + 6C

Now, let's add the two equations:

540 = 12B

Dividing both sides by 12, we get:

B = 45

Now that we have the value of B, we can substitute it into one of the original equations to find the value of C. Let's use the upstream equation:

108 / (B - C) = 3

Substituting B = 45, we get:

108 / (45 - C) = 3

Multiplying both sides by (45 - C), we get:

108 = 3(45 - C)

Expanding the right-hand side, we get:

108 = 135 - 3C

Subtracting 135 from both sides, we get:

-27 = -3C

Dividing both sides by -3, we get:

C = 9

In this article, we used the concept of relative motion to solve a problem involving a motorboat traveling upstream and downstream. We defined the variables involved, formulated the equations, simplified the equations, and solved the equations to find the values of the rate of the boat in still water and the rate of the current. We found that the rate of the boat in still water is 45 km/h and the rate of the current is 9 km/h.

This problem is a classic example of how to apply the concept of relative motion to real-world situations. The concept of relative motion is essential in understanding many phenomena in physics, such as the motion of objects on a moving platform, the motion of a boat in a river, and the motion of a car on a road.

The problem also highlights the importance of defining the variables involved and formulating the equations based on the given information. By simplifying the equations and solving them simultaneously, we can find the values of the variables involved.

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Relative motion: The motion of an object with respect to a reference frame.
Rate of the boat in still water: The speed of the boat in the absence of any external forces, such as wind or current.
Rate of the current: The speed of the water in a river or ocean, which can affect the motion of a boat.
A Motorboat's Journey: Q&A

In our previous article, we explored the concept of relative motion and used it to solve a problem involving a motorboat traveling upstream and downstream. We found that the rate of the boat in still water is 45 km/h and the rate of the current is 9 km/h. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the concept of relative motion?

A: The concept of relative motion refers to the motion of an object with respect to a reference frame. In the context of the motorboat problem, the reference frame is the water in the river or ocean.

Q: Why is it important to define the variables involved in a problem?

A: Defining the variables involved in a problem is crucial because it helps us to formulate the equations that describe the problem. In the motorboat problem, we defined the variables B (rate of the boat in still water), C (rate of the current), D (distance traveled), T (time taken to travel upstream), and T' (time taken to travel downstream).

Q: How do we formulate the equations in a problem?

A: We formulate the equations in a problem by using the given information and the definitions of the variables involved. In the motorboat problem, we used the fact that the boat's speed is reduced due to the opposing force of the current when traveling upstream, and the fact that the boat's speed is increased due to the assisting force of the current when traveling downstream.

Q: What is the difference between the rate of the boat in still water and the rate of the current?

A: The rate of the boat in still water is the speed of the boat in the absence of any external forces, such as wind or current. The rate of the current, on the other hand, is the speed of the water in a river or ocean, which can affect the motion of a boat.

Q: How do we solve the equations in a problem?

A: We solve the equations in a problem by using algebraic techniques, such as substitution and elimination. In the motorboat problem, we used the fact that the distance traveled is the same in both directions to eliminate the variable D.

Q: What is the significance of the motorboat problem?

A: The motorboat problem is a classic example of how to apply the concept of relative motion to real-world situations. It highlights the importance of defining the variables involved and formulating the equations based on the given information.

Q: Can you provide more examples of problems that involve relative motion?

A: Yes, here are a few examples:

A car is traveling at a speed of 60 km/h on a road that is inclined at an angle of 10°. What is the car's speed relative to the road?
A boat is traveling at a speed of 20 km/h in a river that has a current of 5 km/h. What is the boat's speed relative to the shore?
A plane is flying at a speed of 500 km/h in a wind that is blowing at a speed of 50 km/h. What is the plane's speed relative to the ground?

In this article, we answered some frequently asked questions related to the motorboat problem. We hope that this article has provided a better understanding of the concept of relative motion and its application to real-world situations.

The motorboat problem is a classic example of how to apply the concept of relative motion to real-world situations. It highlights the importance of defining the variables involved and formulating the equations based on the given information. We hope that this article has provided a better understanding of the concept of relative motion and its application to real-world situations.

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Relative motion: The motion of an object with respect to a reference frame.
Rate of the boat in still water: The speed of the boat in the absence of any external forces, such as wind or current.
Rate of the current: The speed of the water in a river or ocean, which can affect the motion of a boat.