Question

A man in search of his dog drives first 10.0 mi northeast, then \(12.0 \mathrm{mi}\) straight south, and finally \(8.0 \mathrm{mi}\) in a direction \(30.0^{\circ}\) north of west. What are the magnitude and direction of his resultant displacement?

Short answer

Solution: 1. We break down each displacement into horizontal and vertical components: - 10 miles northeast: \(5\sqrt{2}\) mi east and \(5\sqrt{2}\) mi north - 12 miles south: 0 mi east and -12 mi south - 8 miles 30° north of west: -4 mi west and \(4\sqrt{3}\) mi north 2. We calculate the sum of the horizontal and vertical components: - Horizontal component: \(5\sqrt{2} - 4\) mi - Vertical component: \(5\sqrt{2} - 12 + 4\sqrt{3}\) mi 3. We calculate the magnitude and direction of the resultant displacement: - Magnitude: \(R = \sqrt{(5\sqrt{2} - 4)^2 + (5\sqrt{2} - 12 + 4\sqrt{3})^2}\) - Direction: \(\theta = \arctan{\frac{5\sqrt{2} - 12 + 4\sqrt{3}}{5\sqrt{2} - 4}}\), then convert to degrees The man's resultant displacement is given by its magnitude \(R\) and direction \(\theta\).

Step-by-step solution

Break down each displacement into components

First, we need to break down each displacement into its horizontal (x) and vertical (y) components. Let the displacement towards east be positive in the x-direction, and displacement towards north be positive in the y-direction. For the first displacement (10 mi northeast), this means: \(x_1 = 10 \cos(45) = 10 \cdot \frac{\sqrt{2}}{2} = 5\sqrt{2}\) mi \(y_1 = 10 \sin(45) = 10 \cdot \frac{\sqrt{2}}{2} = 5\sqrt{2}\) mi For the second displacement (12 mi south), there is no horizontal component, and vertical component will be negative: \(x_2 = 0\) mi \(y_2 = -12\) mi For the third displacement (8 mi in direction 30° north of west), we'll have: \(x_3 = 8 \sin(30) = 4\) mi (Note the negative sign as the displacement is towards west) \(y_3 = 8 \cos(30) = 8 \cdot \frac{\sqrt{3}}{2} = 4\sqrt{3}\) mi

Calculate the sum of components

Now we'll calculate the sum of the horizontal components and the sum of the vertical components: \(x = x_1 + x_2 + x_3 = 5\sqrt{2} + 0 - 4 = 5\sqrt{2} - 4\) mi \(y = y_1 + y_2 + y_3 = 5\sqrt{2} - 12 + 4\sqrt{3}\) mi

Calculate the magnitude and direction of the resultant displacement

Now, we can use the Pythagorean theorem to find the magnitude of the resultant displacement: \(R = \sqrt{x^2 + y^2} = \sqrt{(5\sqrt{2} - 4)^2 + (5\sqrt{2} - 12 + 4\sqrt{3})^2}\) Next, we find the direction of the resultant displacement using the arctangent function. We'll first find the angle between the x-axis and the resultant displacement: \(\theta = \arctan{\frac{y}{x}} = \arctan{\frac{5\sqrt{2} - 12 + 4\sqrt{3}}{5\sqrt{2} - 4}}\) Remember to convert the decimal value to degrees by multiplying the result by \(\frac{180}{\pi}\). The final answer will be in the form of the magnitude of the resultant displacement \(R\) and its direction \(\theta\).

Key concepts

Vector Components

Understanding vector components is vital in physics and mathematics to decompose movements into their horizontal and vertical parts. This concept is especially useful when dealing with motions on a plane, like the path of a man in search of his dog.

Imagine walking in a straight line northeast; you are moving in two directions at once: east and north. Decomposing this into vector components means figuring out how much of this movement is purely eastward (the x-component) and how much is purely northward (the y-component).

In our example, the man's first displacement is broken down into two components using trigonometric functions, specifically sine and cosine, which relate the angles of a right triangle to the lengths of its sides. The x-component measures how far he travels east, while the y-component measures how far he travels north. When a displacement is diagonal, both components are typically non-zero. However, a straight south movement has no eastward component, so the x-component is zero.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry linking the three sides of a right triangle. According to this theorem, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In our man's journey, after breaking down each leg of the trip into vector components, we use this theorem to calculate the resultant displacement. By squaring the x and y components and taking the square root of their sum, we find the magnitude of the straight-line distance from the start point to the end point. In effect, this step combines the separate horizontal and vertical motions into a single, direct path.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are crucial tools for analyzing angles and lengths in triangles. They're especially applicable in our scenario for converting angles into actual distances among vector components.

In the provided solution, the calculations for the x and y components relied on the cosine and sine functions. These functions depend on the angle of the displacement to the horizontal. For example, a sine function is used for calculating the y-component, showing how far the man's displacement is vertically, while cosine provides the horizontal x-component. Lastly, the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle, comes into play when determining the angle of displacement from the horizontal axis.

Magnitude and Direction Calculations

The final step in our example involves calculating the magnitude and direction of the resultant displacement. The magnitude is essentially the 'length' of the displacement vector and is found using the Pythagorean Theorem on the respective components, as previously described.

As for direction, it's determined by calculating the angle the resultant vector makes with the horizontal axis. This angle is computed using the arctangent function, which is particularly used when the lengths of the opposite and adjacent sides are known but the angle isn't. Once we have the angle in radians, it is often converted to degrees since degrees are more intuitive for most people to understand. By combining the magnitude and angle, we get a complete picture of the man's overall displacement in a single vector.