Problem 33 - University Physics with Modern Physics

Question

A spelumker is surveying a cave. She 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 \(30^{\circ}\) east of north. After a fourth unmeasured displacement, she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth displacement. (See also Problem 1.73 for a different approach to this problem.)

Step-by-Step Solution

Answer

The fourth displacement is 143.44 m at an angle of 40.15° south of west.

Step 1: Analyze the First Movement

The spelunker travels 180 m straight west. In our coordinate system, west corresponds to the negative x-axis. Thus, the displacement vector for this movement is \((-180 \text{ m}, 0 \text{ m})\).

Step 2: Analyze the Second Movement

The next movement is 210 m at an angle of \(45^{\circ}\) east of south. We can break this down into components using trigonometry:- Southward (negative y-direction): \(210 \cos(45^{\circ}) = 148.5 \text{ m}\)- Eastward (positive x-direction): \(210 \sin(45^{\circ}) = 148.5 \text{ m}\)This results in the displacement vector \((148.5 \text{ m}, -148.5 \text{ m})\).

Step 3: Analyze the Third Movement

The spelunker then travels 280 m at \(30^{\circ}\) east of north. Similarly, we find the components:- Northward (positive y-direction): \(280 \cos(30^{\circ}) = 242.49 \text{ m}\)- Eastward (positive x-direction): \(280 \sin(30^{\circ}) = 140 \text{ m}\)Thus, the displacement vector is \((140 \text{ m}, 242.49 \text{ m})\).

Step 4: Sum the Vectors

Add all the displacement vectors to find the resultant vector. This will give the total position from the start before the fourth displacement:- Total x-component: \(-180 + 148.5 + 140 = 108.5 \text{ m}\) (east)- Total y-component: \(0 - 148.5 + 242.49 = 93.99 \text{ m}\) (north)The result is \((108.5, 93.99)\).

Step 5: Calculate the Fourth Displacement

Since the spelunker returns to her starting point, the fourth displacement must negate the resultant vector from Step 4. Therefore, the fourth vector is \((-108.5, -93.99)\). Convert these into magnitude and direction:- Magnitude: \(\sqrt{(-108.5)^2 + (-93.99)^2} \approx 143.44 \text{ m}\)- Direction: \(\tan^{-1}\left(\frac{-93.99}{-108.5}\right) \approx 40.15^{\circ}\) south of west.

Key Concepts

Displacement Vectors

In the context of vector addition, displacement vectors are crucial in describing movement across a plane. A displacement vector details both the magnitude and direction from one point to another. When a spelunker or any other individual is navigating a space, these vectors help track their precise location changes. Displacement vectors are useful because they give a complete picture of movement without needing to know the path taken between the start and end points. Each movement or leg of the journey can be represented as a vector with specific components along an axis, like moving west, east, north, or south in a cave system. By combining these vectors, we obtain a resultant vector representing the total change in position. If several movements eventually bring you back to your starting point, the resultant vector would be nullified by the final movement, effectively balancing it out.

Trigonometry

Trigonometry is indispensable in breaking down vectors into their components, especially when the vectors are not aligned with the main axes. Components are the projections of the vector along the x and y axes and are found using trigonometric functions like sine and cosine.For example, when the spelunker moves 210 m at an angle of 45° east of south, we can use trigonometry to break this movement into:\[\text{South component (y)} = 210 \times \cos(45^{\circ}) \approx 148.5 \, \text{m} \\text{East component (x)} = 210 \times \sin(45^{\circ}) \approx 148.5 \, \text{m} \]This process is repeated with each movement, turning vectors into manageable parts that can be easily summed. By mastering trigonometry, you can interpret, decompose, and reassemble vectors in various directions, making it easier to solve complex spatial problems.

Scale Drawing

Scale drawings are an essential tool for visualizing problems involving vectors, especially when dealing with directions and angles. By using a consistent scale, you can create a scaled-down version of the problem, making it easier to understand and solve. A scale drawing allows you to represent each vector's direction and magnitude visually. For example, using a paper to draw the vectors the spelunker followed allows you to lay out the cave survey. These drawings help you estimate the final displacement needed to return to the starting point. While computing the final answer mathematically is precise, a scale drawing provides intuitive insight and can make complex vector problems feel more tangible. When vectors are plotted accurately, the fourth vector—which returns the spelunker to her starting point—becomes apparent by simply measuring on the drawing and using it for a more visual confirmation.