Question

The water in a river moves south at \(5 \mathrm{km} / \mathrm{hr} .\) A motorboat travels due east at a speed of \(40 \mathrm{km} / \mathrm{hr}\) relative to the water. Determine the speed of the boat relative to the shore.

Short answer

Answer: The speed of the motorboat relative to the shore is approximately 40.31 km/hr.

Step-by-step solution

Identify the given information

We are given the following information: - The velocity of the water in the river is 5 km/hr (south) - The velocity of the motorboat relative to the water is 40 km/hr (east)

Represent the given velocities as vectors

We can represent the velocities as vectors: - Velocity of water: \(\vec{v}_{w} = 5\,\text{km/hr}\, \hat{j}\) - Velocity of motorboat relative to the water: \(\vec{v}_{bw} = 40\,\text{km/hr}\, \hat{i}\)

Find the velocity of the motorboat relative to the shore

To find the velocity of the motorboat relative to the shore, we need to add the velocity vectors of the boat relative to the water and the velocity of the water: $$\vec{v}_{bs} = \vec{v}_{bw} + \vec{v}_{w}$$ $$\vec{v}_{bs} = 40\,\text{km/hr}\, \hat{i} + 5\,\text{km/hr}\, \hat{j}$$

Calculate the magnitude of the resultant vector

Now, we need to determine the speed of the boat relative to the shore by finding the magnitude of the resultant vector: $$|\vec{v}_{bs}| = \sqrt{(40\,\text{km/hr})^2 + (5\,\text{km/hr})^2}$$ $$|\vec{v}_{bs}| = \sqrt{1600\,\text{km}^2/\text{hr}^2 + 25\,\text{km}^2/\text{hr}^2}$$ $$|\vec{v}_{bs}| = \sqrt{1625\,\text{km}^2/\text{hr}^2}$$ $$|\vec{v}_{bs}| = 40.31\,\text{km/hr}$$

Final answer

The speed of the motorboat relative to the shore is approximately \(40.31\,\text{km/hr}\).

Key concepts

Relative Velocity

When we discuss relative velocity, we are looking at the velocity of an object in relation to the velocity of another object. It's an essential concept in physics, especially when objects move within mediums that also have their own movement, like boats in water or planes in air.

For instance, in the exercise, we examine a motorboat going east and water moving south. The motorboat's velocity is given in relation to the water, but we want to find the boat’s velocity relative to the shore, which is stationary. To solve this, we apply vector addition, which allows us to combine the velocity of the water and the velocity of the boat to obtain the relative velocity of the boat with respect to the shore. This calculation provides us with the actual path and speed the motorboat will follow from an observer's point of view on the shore, taking the water's current into account.

Magnitude of a Vector

The magnitude of a vector represents the size or length of the vector. It's a scalar quantity, meaning it has a magnitude but no direction. In our context, it signifies the speed of the motorboat regardless of the direction.

Let's look at the method to calculate the magnitude using the Pythagorean theorem. The velocities of the water and the boat form a right-angled triangle with the velocities as sides. Calculating the magnitude is like finding the hypotenuse of this triangle, which gives us the true speed of the boat relative to the land. The formula used is the square root of the sum of the squares of the individual vector components, which can be mathematically represented as:
\(|\text{Magnitude}| = \[\sqrt{(\text{vector}_{x})^2 + (\text{vector}_{y})^2}\]\).

In our exercise, the magnitude, which is the speed of the boat relative to the shore, is approximately 40.31 km/hr, found through this calculation.

Vector Representation

The term vector representation refers to the way we express vectors, which include both direction and magnitude. The vectors are typically illustrated with arrows, where the direction of the arrow represents the direction of the vector and the length represents its magnitude.

In our textbook example, the velocity of the water and the motorboat relative to the water are represented as vectors \(\vec{v}_{w} = 5\,\text{km/hr}\, \hat{j}\) and \(\vec{v}_{bw} = 40\,\text{km/hr}\, \hat{i}\). Here, \(\hat{i}\) and \(\hat{j}\) are unit vector representations pointing in the east and south directions, respectively. These notations are part of a coordinate system which helps in simplifying the addition or subtraction of vectors, among other operations. This approach allows us to easily visualize and calculate the impact of different movements when they happen concurrently, which is key to solving problems involving relative motion.