Problem 53 - College Physics

Question

While surveying a cave, a spelunker follows a passage \(180 \mathrm{~m}\) straight west, then \(210 \mathrm{~m}\) in a direction \(45^{\circ}\) east of south, and then \(280 \mathrm{~m}\) at \(30.0^{\circ}\) east of north. After a fourth unmeasured displacement, she finds herself back where she started. Use vector components to find the magnitude and direction of the fourth displacement. Then check the reasonableness of your answer with a graphical sum.

Step-by-Step Solution

Answer

The fourth displacement is approximately 211.2 m at 2.3° east of south.

Step 1: Define the Problem

The spelunker moves in three known displacements and one unknown displacement. Our goal is to calculate the magnitude and direction of this fourth unknown displacement vector, denoted as \( \mathbf{D}_4 \). By vector addition, the sum of all displacements should equal zero since she returns to the starting point: \( \mathbf{D}_1 + \mathbf{D}_2 + \mathbf{D}_3 + \mathbf{D}_4 = 0 \).

Step 2: Resolve Vector Components for Each Displacement

Each vector is resolved into its north-south and east-west components. The angles are measured from specified directions.- **\( \mathbf{D}_1 \):** Moves west, so its components are: \( D_{1x} = -180 \mathrm{~m} \), \( D_{1y} = 0 \mathrm{~m} \).- **\( \mathbf{D}_2 \):** Moves \(45^{\circ}\) east of south: \( D_{2x} = 210 \cos(45^{\circ}) \mathrm{~m}, D_{2y} = -210 \sin(45^{\circ}) \mathrm{~m} \).- **\( \mathbf{D}_3 \):** Moves \(30^{\circ}\) east of north: \( D_{3x} = 280 \cos(30^{\circ}) \mathrm{~m}, D_{3y} = 280 \sin(30^{\circ}) \mathrm{~m} \).

Step 3: Calculate Vector Components

Using the angle information and trigonometric functions:- **\( \mathbf{D}_2 \):** \( D_{2x} = 210 \cdot \frac{\sqrt{2}}{2} \approx 148.5 \mathrm{~m}, D_{2y} = -210 \cdot \frac{\sqrt{2}}{2} \approx -148.5 \mathrm{~m} \).- **\( \mathbf{D}_3 \):** \( D_{3x} = 280 \cdot \cos(30^{\circ}) = 280 \cdot \frac{\sqrt{3}}{2} \approx 242.5 \mathrm{~m}, D_{3y} = 280 \cdot \sin(30^{\circ}) = 280 \cdot \frac{1}{2} = 140 \mathrm{~m} \).

Step 4: Sum Components for Total Displacement

Add the components of each vector to find the total displacement vector:\( D_{x, \text{total}} = D_{1x} + D_{2x} + D_{3x} = -180 + 148.5 + 242.5 \approx 211 \mathrm{~m} \).\( D_{y, \text{total}} = D_{1y} + D_{2y} + D_{3y} = 0 - 148.5 + 140 \approx -8.5 \mathrm{~m} \).

Step 5: Find Fourth Displacement Vector

Since the total displacement must be zero, the fourth displacement \( \mathbf{D}_4 \) has components:\( D_{4x} = -D_{x, \text{total}} = -211 \mathrm{~m} \), \( D_{4y} = -D_{y, \text{total}} = 8.5 \mathrm{~m} \).

Step 6: Calculate Magnitude and Direction of \( \mathbf{D}_4 \)

The magnitude of \( \mathbf{D}_4 \) is calculated using the Pythagorean theorem:\[ |\mathbf{D}_4| = \sqrt{D_{4x}^2 + D_{4y}^2} = \sqrt{(-211)^2 + (8.5)^2} \approx 211.2 \mathrm{~m} \].The direction \( \theta \), east of south, is given by:\[ \theta = \tan^{-1}\left(\frac{8.5}{211}\right) \approx 2.3^{\circ} \].

Step 7: Verify with Graphical Method

Sketch the vectors from start to end. The graphical addition should form a closed shape with a resultant vector pointing in the opposite direction to \(\mathbf{D}_4\). The fourth vector closes the shape accurately indicating the calculated solution is reasonable.

Key Concepts

Displacement

Displacement is a fundamental concept in physics and describes the shortest straight-line distance from a starting point to an endpoint. Unlike distance, which considers the entire path length traveled, displacement only focuses on the initial and final positions.
In the described exercise, the spelucker's journey consists of multiple displacements. Each movement in a specific direction adds up to the total displacement. Since she ends up back at her starting position, her total displacement for the round trip is zero. This is critical when considering the need for calculating the unknown fourth displacement, as each displacement vector sums to zero together.
Understanding displacement helps simplify the process of resolving complex paths into manageable vectors, making it easier to compute the overall journey.

Vector Components

Vectors can often be simplified into their components, typically in north-south or east-west directions. Breaking them down like this makes it easier to analyze and calculate their effects.
A component is essentially a projection of a vector along the axes—usually the x (horizontal) and y (vertical) axes. Each movement of the spelunker is represented by these components. For example, a movement west would solely influence the x-component, or east-west movement, of the vector.
By breaking down every displacement into its vector components, we arrive at a simple sum for those in the east-west direction and those in the north-south direction.

• East-West Components: Influenced by movements purely in that axis direction.
• North-South Components: Affected by vectors that have a movement component in the vertical direction.

This breakdown is essential for solving for the final, unknown displacement of the journey.

Vector Resolution

Vector resolution is the process of breaking a vector into its components. It involves using trigonometric functions based on the direction's angle, which helps in determining the exact influence of a vector along specified axes.
For instance, a vector oriented at an angle requires the use of cosine and sine functions to resolve it into its respective horizontal and vertical parts, otherwise known as components. In the current exercise, the spelunker's movements at varying angles require vector resolution.
When resolving vectors:

• For angles measured from horizontal/vertical lines, use cosine for the horizontal component and sine for the vertical component.
• For complex movements, application of the \(\cos\) and \(\sin\) functions helps derive accurate vector components.

This method ensures precise measurement, leading to accurate summation of all vector components.

Graphical Method

The graphical method provides a visual verification tool to ensure calculations are reasonable and correct. This approach involves drawing each vector from start to finish, maintaining scale and direction accurately.
In the exercise scenario, after calculating the numerical solution, sketch the known vectors in succession. The intention is to form a completed loop, indicative of returning to the start due to the closed path formed by vectors.
Steps to use the graphical method:

• Draw vectors to scale, based on magnitude and direction.
• Connect the vectors tip-to-tail from start to end.
• The resultant vector should close the shape back to the original point if correct.

Using this graphical method supports numerical calculation integrity by visually verifying the spatial arrangement of vector yield.