Problem 78 - University Physics with Modern Physics

Question

Getting Back. An explorer in the dense jungles of equatorial Africa leaves his hut. He takes 40 steps northeast, then 80 steps \(60^{\circ}\) north of west, then 50 steps due south. Assume his steps all have equal length. (a) Sketch, roughly to scale, the three vectors and their resultant, (b) Save the explorer from becoming hopelessly lost in the jungle by giving him the displacement, calculated using the method of components, that will return him to his hut.

Step-by-Step Solution

Answer

Move 11.72 steps east and 47.56 steps south to return to the hut.

Step 1: Understand the Problem as Vector Components

We need to find the vector displacement that returns the explorer to his original starting point. We will break down each movement into components along the x-axis (east-west) and y-axis (north-south).

Step 2: Determine Components of Northeast Movement

The first movement is 40 steps northeast, which can be resolved into components along the x and y axes. Since northeast is at a 45° angle from the axes, we use \[40 \times \cos(45°) = 40 \times \sin(45°) = 40 \times \frac{\sqrt{2}}{2} \approx 28.28.\] Thus, this movement is approximately 28.28 steps to the east (positive x) and 28.28 steps to the north (positive y).

Step 3: Calculate Components of 60° North of West Movement

The second movement is 80 steps at 60° north of west. For the x component (west), we have \[80 \times \cos(60°) = 80 \times \frac{1}{2} = 40.\] For the y component (north), \[80 \times \sin(60°) = 80 \times \frac{\sqrt{3}}{2} \approx 69.28.\] Therefore, the explorer moves 40 steps to the west (negative x) and approximately 69.28 steps to the north (positive y).

Step 4: Determine Components of Due South Movement

The last movement is 50 steps due south, which only affects the y component. Specifically, this is \[-50\] steps in the y direction (south). There is no change in the x direction.

Step 5: Sum All Component Vectors

To find the total displacement, we sum up the x and y components separately. For x-components: \[28.28 - 40 = -11.72.\] For y-components: \[28.28 + 69.28 - 50 = 47.56.\] Thus, the resultant vector is \([-11.72, 47.56].\)

Step 6: Calculate the Opposite Vector Required

To return to the original location, we need the opposite of the resultant displacement vector determined in Step 5. Therefore, the displacement to return is \([11.72, -47.56].\) This means the explorer should move approximately 11.72 steps east and 47.56 steps south.

Key Concepts

Vector Addition

Vector addition is a fundamental concept in physics and mathematics. It helps us find the resultant of multiple vectors combined. When vectors are represented on a plane, they can be broken down into components along the x-axis and y-axis. To add vectors, you simply add all the x components together for the resultant x, and all the y components together for the resultant y.

In our explorer's journey, we start by breaking each movement into its x and y directions. For his first movement northeast, the vector's x and y parts are the same due to the 45° angle to both axes, resulting in equal east and north components. The second vector, directed 60° north of west, shows a westward (negative x) and northward (positive y) component—both calculated using sine and cosine functions. The final movement, due south, has only a y component, going in negative direction.

This approach allows us to add the x parts together and the y parts together, simplifying multidimensional problems into easy scalar additions. Once summed, these provide a new vector giving the overall change in position. By reversing this resultant vector, the explorer could head back to his starting point.

Displacement Calculation

Calculating displacement is about finding the shortest distance and direction between two points. Displacement is a vector quantity, meaning it has both magnitude and direction. We find it using vector components in our exercise example to save the explorer from getting lost in the jungle.

Once we resolve each movement into components and calculated their sums, we get a resultant vector that shows overall displacement. For our explorer, this is approximately [-11.72, 47.56] steps. This tells us the explorer is located 11.72 steps to the west and 47.56 steps to the north of his starting point. To find the direction back home, he needs to take the opposite displacement: 11.72 steps to the east and 47.56 steps south.

This systematic way of finding displacement helps ensure accuracy in navigation since even small errors in wilderness navigation can be significant.

Trigonometry in Physics

Trigonometry plays a crucial role in vector analysis, helping break down vectors into components. It leverages sine and cosine functions to resolve a vector at an angle into its horizontal and vertical components.

In the exercise, trigonometric functions are key when determining components of the explorer's path. Walking 40 steps northeast involves trigonometry as each component equals the step size multiplied by \( \cos(45°) \) or \( \sin(45°) \), due to the 45-degree angle. Similar logic is applied for the movement 60° north of west. Here, the cosine of 60° points to the x component, and the sine to the y component.

Understanding these concepts simplifies the separation of vectors into directly manageable parts that reflect real-world movements and angles. This is invaluable not just in physics classes but also in practical applications like navigating environments.