Problem 68 - University Physics with Modern Physics

Question

An explorer in Antarctica leaves his shelter during a whiteout. He takes 40 steps northeast, next 80 steps at 60\(^{\circ}\) north of west, and then 50 steps due south. Assume all of his steps are equal in length. (a) Sketch, roughly to scale, the three vectors and their resultant. (b) Save the explorer from becoming hopelessly lost by giving him the displacement, calculated by using the method of components, that will return him to his shelter.

Step-by-Step Solution

Answer

The explorer should take steps opposite to the resultant vector to return.

Step 1: Define Vectors for Each Leg of the Journey

For each segment of the explorer's path, we will assign a vector. Let \( \vec{A} \) be the vector representing his path going 40 steps northeast. Northeast implies a 45-degree angle between the north and east axes. Vector \( \vec{B} \) represents his path going 80 steps at 60° north of west, and vector \( \vec{C} \) represents his path going 50 steps due south.

Step 2: Calculate Components of Vector A

To find the components of vector \( \vec{A} \) (northeast), we must remember that northeast means 45 degrees from both north and east. Therefore, the x-component (east direction) and y-component (north direction) are both \( 40 \times \cos(45^{\circ}) \) and \( 40 \times \sin(45^{\circ}) \) respectively. Calculate these as:\[A_x = 40 \times \cos(45^{\circ}) = 40 \times \frac{\sqrt{2}}{2} = 20\sqrt{2} \A_y = 40 \times \sin(45^{\circ}) = 40 \times \frac{\sqrt{2}}{2} = 20\sqrt{2}\]

Step 3: Calculate Components of Vector B

Vector \( \vec{B} \) has a magnitude of 80 steps and is directed 60° north of west. To find its components, we use west as negative x-direction and north as positive y-direction. Therefore:\[B_x = 80 \times \cos(120^{\circ}) = 80 \times (-\frac{1}{2}) = -40 \B_y = 80 \times \sin(120^{\circ}) = 80 \times \frac{\sqrt{3}}{2} = 40\sqrt{3}\]

Step 4: Calculate Components of Vector C

Vector \( \vec{C} \) is purely in the south direction. For a step length of 50, this means:\[C_x = 0, \quad C_y = -50\]

Step 5: Sum the Components of All Vectors

Now, sum up all the x and y components of vectors \( \vec{A}, \vec{B}, \vec{C} \) to find the resultant vector \( \vec{R} \):\[R_x = A_x + B_x + C_x = 20\sqrt{2} - 40 + 0 \R_y = A_y + B_y + C_y = 20\sqrt{2} + 40\sqrt{3} - 50\]

Step 6: Calculate Displacement for Return

To determine the displacement needed to return, we calculate the vector \( \vec{R} \' \) that is opposite to \( \vec{R} \):\[-R_x = -(20\sqrt{2} - 40) \-R_y = -(20\sqrt{2} + 40\sqrt{3} - 50)\] This final vector \( \vec{R} \' \) gives the displacement for the explorer to return to the shelter.

Key Concepts

Displacement Calculation

Displacement is a straightforward physics concept often represented as a vector quantity. It signifies a change in position and is expressed with both a magnitude and direction. When calculating displacement, especially in multi-directional travels like this explorer's journey, it's crucial to consider all the displacement vectors in play.

In this problem, our poor wanderer starts getting off track the moment he steps away from his shelter in different directions. Displacement calculation not only establishes his total journey in physical distance but also computes the direction needed for him to return to his starting point. By sketching his path and finding the resultant vector that combines the three given vectors, we determine the exact displacement he needs to rectify his course.

Remember that displacement is different from distance; while distance adds up all the steps without considering the direction, displacement focuses on the shortest straight path between start and end point. This ensures our explorer finds the most efficient route back to his shelter.

Component Method

The component method is an efficient way to solve vector problems by simplifying them into basic horizontal and vertical components. It transforms a seemingly complicated vector into more manageable parts, allowing easier computation using basic trigonometry.

Here’s how it works in the context of the explorer's wander:

• Vector A is directed northeast, thus balanced equally between the east and north. Its components are found using angles, typically 45 degrees, due to northeast directionality.
• Vector B travels north of west. We must convert this into components along the traditional X (west/east) and Y (north/south) axes. The angle here is vital because it affects both the direction and the sign of the components (negative for west).
• Vector C is purely in the Y-axis direction going south, making its computation straightforward as there’s no X component.

By breaking down each vector into x and y components, calculating them using sine and cosine based on respective angles like 45°or 120°, and summing them up, this method grants clarity in determining the final, resultant displacement.

Physics Problem Solving

Solving physics problems is all about understanding the concepts and applying them effectively. This particular problem, while seemingly a straightforward navigation exercise, allows us to delve into broader physics problem-solving techniques.

First, it emphasizes the importance of visualization. Sketching the vectors provides an immediate sense of the directions and magnitudes involved. By visualizing the journey, one can more readily grasp the context and forming a reliable path toward the solution.

Next, transformation is key. Using the component method, a complex vector problem becomes imminently more accessible, forming the backbone of strategy in complex physics scenarios. The breakdown into x and y components is not only vital for calculating displacements but also for learning how different forces coalesce in various directions.

Lastly, one must always check their calculations. Physics problems often seem tricky not because the concepts are difficult, but because meticulous attention to detail is required. Ensuring each calculation follows logically and incorporates every component is crucial in finding the right solution. Maintaining a calm and methodical approach helps rescue not just explorers, but also students wrestling with challenging physics problems.