Problem 61 - 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

Magnitude and direction calculated will verify the spelunker's fourth displacement completes the circular route.

Step 1: Break Down Each Displacement into Components

To start, break down each of the given displacement vectors into their x (east-west) and y (north-south) components using trigonometry. The directions and lengths of the paths need careful attention. For the 180 m straight west: \( x_1 = -180 \) m (since west is negative x-direction), \( y_1 = 0 \) m. For 210 m at \( 45^{\circ} \) east of south, \( x_2 = 210 \sin 45^{\circ} \), \( y_2 = -210 \cos 45^{\circ} \) because south is the negative y-direction. For 280 m at \( 30.0^{\circ} \) east of north, \( x_3 = 280 \sin 30^{\circ} \), \( y_3 = 280 \cos 30^{\circ} \) because north is positive y-direction.

Step 2: Calculate Total x and y Displacement Components

Sum all the x components and y components separately. \( x_{total} = x_1 + x_2 + x_3 \) and \( y_{total} = y_1 + y_2 + y_3 \). This will give you the resultant vectors from the initial displacements that lead to the final position being where the spelunker started.

Step 3: Determine the Fourth Displacement Components

Since the spelunker ends up at the starting point, the fourth displacement must cancel out the total displacement calculated in Step 2. Thus, the components for the fourth displacement, \( x_4 \) and \( y_4 \), need to be \( x_4 = -x_{total} \) and \( y_4 = -y_{total} \).

Step 4: Calculate Magnitude of the Fourth Displacement

Use the Pythagorean theorem to find the magnitude of the fourth displacement: \( d_4 = \sqrt{(x_4)^2 + (y_4)^2} \). This provides the length of the vector.

Step 5: Determine Direction of the Fourth Displacement

Use the tangent inverse function to determine the direction of the fourth displacement relative to the positive x-axis: \( \theta_4 = \tan^{-1}(\frac{y_4}{x_4}) \). This gives you the angle the displacement vector makes.

Step 6: Verify Solution Graphically

Sketch the vectors based on your previous calculations. Ensure that the graphical vector addition of paths forms a closed shape, returning to the origin, to visualize the correctness of your components and magnitude.

Key Concepts

Displacement Vectors

Displacement vectors represent the change in position of an object from its initial point to its final point in a straight line. In this context, we're looking at a spelunker, or cave explorer, who travels along different paths inside a cave. Each path represents a vector with a specific direction and magnitude. For example, moving 180 meters straight west forms a vector pointing entirely towards the west. Each vector tells you exactly how far and in what direction the spelunker has moved.

To fully understand a displacement vector, you need to know both its magnitude and direction:

• Magnitude: This refers to the length of the vector, showing how far the object travels.
• Direction: This is the angle or bearing the vector is pointing towards, which in this problem involves directions like straight west or east of north.

Breaking vectors into components helps manage these directional details more easily, especially when dealing with complex paths in trigonometry.

Trigonometry

Trigonometry, the study of the relationships between angles and lengths in triangles, is a key tool for working with displacement vectors. When you know the direction and distance of travel (the vector's magnitude and direction), trigonometric functions can help break each vector into x (east-west) and y (north-south) components. This makes it easier to solve problems involving multiple vectors.

For example, if a vector's direction forms a 45° angle east of south, you can use the sine and cosine functions:

• To find the x-component, calculate: \( x = ext{{magnitude}} imes ext{{sin}}( ext{{angle}}) \)
• To find the y-component, calculate: \( y = ext{{magnitude}} imes ext{{cos}}( ext{{angle}}) \)

These calculations allow you to express each movement in Cartesian coordinates, which simplify the process of adding multiple vectors together.

Magnitude and Direction

Every vector has both a magnitude, which tells you how long it is, and a direction, which indicates where it points. Understanding these two components is crucial for solving vector-related problems.

Magnitude can be found using the Pythagorean theorem if you know the vector's x and y components:\[m = \ \sqrt{x^2 + y^2}\]This formula provides the vector's length. In practical applications, like the spelunker's journey, finding the magnitude of the resulting vector can confirm the total change in position.

Direction, on the other hand, is determined using trigonometry. If you know a vector's x and y components, you can find its direction as an angle from the positive x-axis with the inverse tangent function:\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]This direction, when combined with the magnitude, gives a complete description of the vector's position.

Graphical Vector Addition

Graphical vector addition helps visualize how multiple vectors combine to form a resultant vector. This process involves arranging the vectors head to tail in a diagram to see how they form a path.

In the spelunker's cave survey, each leg of her journey can be represented as a vector on a simple map:

• First, draw the first vector from the starting point.
• Then, start the next vector where the last one ended, following its direction and magnitude.
• Continue this process for each vector in sequence.

The final vector should connect the end of the last movement back to the start, ensuring the vector forms a closed shape, effectively visualizing that the spelunker returns to her starting point. This graphical check confirms whether the mathematical calculations align with the physical journey inside the cave.