Question

During a jaunt on your sailboat, you sail \(2.00 \mathrm{~km}\) east, then \(4.00 \mathrm{~km}\) southeast, and finally an additional distance in an unknown direction. Your final position is \(6.00 \mathrm{~km}\) directly east of the starting point. Find the magnitude and direction of the third leg of your journey.

Short answer

In the given scenario, the third leg of the journey has a magnitude of approximately 3.54 km, and the direction is 23.2° southeast from the final position.

Step-by-step solution

Determine the initial position and the final position as vectors

First, we assign the initial position vector as \(\mathbf{O}\). Then, after sailing \(2.00 \mathrm{~km}\) east, we have the position vector \(\mathbf{A} = 2.00\mathbf{i}\). Next, after sailing \(4.00 \mathrm{~km}\) southeast, we have the position vector \(\mathbf{B}\). Since southeast is \(45^\circ\) from east, we can express \(\mathbf{B}\) as: \(\mathbf{B} = 2.00\mathbf{i} + 4.00(\cos{45^\circ}\mathbf{i} + \sin{45^\circ}\mathbf{j})\). Finally, the final position is \(6.00\mathrm{~km}\) east of the starting point, so the position vector is \(\mathbf{F} = 6.00\mathbf{i}\).

Calculate the third leg vector

To find the vector representing the last part of the journey, we will subtract the initial position vector \(\mathbf{O}\) from the final position vector \(\mathbf{F}\). This will give us the vector connecting the third leg of the journey. Let's denote the final vector as \(\mathbf{T}\): \(\mathbf{T} = \mathbf{F} - \mathbf{B} = 6.00\mathbf{i} - (2.00\mathbf{i} + 4.00(\cos{45^\circ}\mathbf{i} + \sin{45^\circ}\mathbf{j}))\). Calculating this gives: \(\mathbf{T} = (6.00 - 2.00 - 4.00\cos{45^\circ})\mathbf{i} - 4.00\sin{45^\circ}\mathbf{j}\).

Find the magnitude and direction of the third leg vector

We now have the components of the third leg vector, so we can use the Pythagorean theorem to find its magnitude. The magnitude of \(\mathbf{T}\) is given by \(|\mathbf{T}| = \sqrt{((6.00 - 2.00 - 4.00\cos{45^\circ})^2 + (4.00\sin{45^\circ})^2)}\). Evaluate this to obtain: \(|\mathbf{T}| \approx 3.54 \mathrm{~km}\). To find the direction of the third leg vector, we can use the arctangent function to find the angle between the horizontal component and the vertical component of \(\mathbf{T}\). The angle \(\theta\) is given by: \(\theta = \arctan{(\frac{4.00\sin{45^\circ}}{6.00 - 2.00 - 4.00\cos{45^\circ}})}\). Evaluating this gives us approximately: \(\theta \approx 23.2^\circ\). Since the horizontal component is positive and the vertical component is negative, the direction is southeast from the final position.

Answer

The magnitude of the third leg of the journey is approximately \(3.54 \mathrm{~km}\), and the direction is \(23.2^\circ\) southeast from the final position.

Key concepts

Vector Components

Understanding vector components is essential to solve many physics problems, especially ones involving motion. A vector represents both magnitude and direction, making it more informative than just a scalar, which only has magnitude. When we talk about vector components, we're referring to breaking down a vector into its parts along the coordinate axes, usually represented as the x-component (horizontal) and y-component (vertical).

For example, in the sailboat problem, the second leg of the journey is southeast, which means that the vector can be broken down into horizontal (eastward) and vertical (southward) components. Using trigonometry, we find that the eastward (x-component) is given by the cosine of the angle times the magnitude, and the southward (y-component) by the sine of the angle times the magnitude. In this way, we can pinpoint the boat’s change in position on a map.

Magnitude and Direction Calculation

Calculating the magnitude and direction is a crucial step when dealing with vectors, as it gives us concrete information about how far and in which direction an object has moved. The magnitude of a vector (often denoted as |vector| or simply 'magnitude') refers to its size or length and can be calculated by using the Pythagorean theorem when we know the vector’s components.

In the context of our sailing problem, once we have identified the vector components for the third leg of the journey, we use the Pythagorean theorem to calculate the magnitude. This process involves squaring both components, adding these squares together, and taking the square root of the sum. The direction, on the other hand, is determined by calculating the angle the vector makes with a reference axis, using trigonometric functions such as arctangent.

Pythagorean Theorem in Physics

The Pythagorean theorem is not just a concept learned in math class; it's also widely used in physics, particularly in problems involving right triangles and vector analysis. According to this theorem, in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

For our sailing scenario, this theorem helps us find the magnitude of the third leg of the journey – the hypotenuse of the triangle formed by vector components. We calculate the magnitude by squaring the horizontal and vertical components, adding these together, and then taking the square root of this sum. This method provides a straightforward way to move from the vector components back to the overall vector magnitude, which represents the actual distance sailed during the third leg of the journey.

Arctangent Function for Angle

Determining the angle between the vector’s path and a reference line is often necessary to fully understand a vector’s direction. This is where trigonometry, and specifically the arctangent function, comes into play. The arctangent (or inverse tangent) function allows us to calculate the angle of a vector from its components. Essentially, it gives the angle whose tangent is the ratio of the vector's y-component to its x-component.

In our sailboat problem, after finding the components of the third leg’s vector, we use the arctangent function to find the angle of original direction of travel, by inputting the ratio of the vertical component over the horizontal component as the argument. Understanding this concept is vital, as it translates the technical vector component information into a literal direction that can be visualized on a compass or a map – such as '23.2 degrees southeast'.