Question

While following a treasure map, you start at an old oak tree. You first walk 825 m directly south, then turn and walk 1.25 km at 30.0\(^{\circ}\) west of north, and finally walk 1.00 km at 32.0\(^{\circ}\) north of east, where you find the treasure: a biography of Isaac Newton! (a) To return to the old oak tree, in what direction should you head and how far will you walk? Use components to solve this problem. (b) To see whether your calculation in part (a) is reasonable, compare it with a graphical solution drawn roughly to scale.

Short answer

Head approximately 750 m directly back to the south-east, adjusting angle from north accordingly.

Step-by-step solution

Convert Distances to Same Units

Convert all distances to the same unit for consistency:1. The first walk is already given in meters: 825 m.2. The second walk is 1.25 km. Convert to meters: \[ 1.25 \text{ km} = 1.25 \times 1000 = 1250 \text{ m} \]3. The third walk is 1.00 km. Convert to meters: \[ 1.00 \text{ km} = 1.00 \times 1000 = 1000 \text{ m} \]Now, all distances are in meters: 825 m, 1250 m, and 1000 m.

Break Down Movements into Components

Use trigonometry to break each movement into its north-south and east-west components.- For the first movement (825 m south), the components are: - North-South: -825 m (since it's directly south, it's negative) - East-West: 0 m (no movement in this direction)- For the second movement (1250 m, 30.0° west of north): - North-South: \( 1250 \times \cos(30.0^{\circ}) \) - East-West: \( -1250 \times \sin(30.0^{\circ}) \) (negative due to westward direction)- For the third movement (1000 m, 32.0° north of east): - North-South: \( 1000 \times \sin(32.0^{\circ}) \) - East-West: \( 1000 \times \cos(32.0^{\circ}) \)

Calculate Total Components

Sum all the north-south and east-west components to find the total displacement.- North-South component: \[ -825 + 1250 \times \cos(30.0^{\circ}) + 1000 \times \sin(32.0^{\circ}) \]- East-West component: \[ 0 - 1250 \times \sin(30.0^{\circ}) + 1000 \times \cos(32.0^{\circ}) \]Compute these using a calculator.

Determine Return Vector

The return vector is the negative of the total displacement. Calculate the magnitude and direction.- Magnitude of the resultant vector: \[ \sqrt{(\text{North-South component})^2 + (\text{East-West component})^2} \]- Direction (angle \( \theta \)) from the North towards East: \[ \theta = \tan^{-1}\left(\frac{\text{East-West component}}{\text{North-South component}}\right) \]Adjust the angle based on the quadrant.

Verify with Graphical Representation

Draw a rough scale diagram of the path: 1. Use a ruler and protractor with appropriate scale. 2. Trace the paths as per the given distances and angles. 3. Directly draw the calculated return vector. Ensure it visually coincides with computed distance and heading. This acts as a double-check for direction and distance consistency.

Key concepts

Trigonometry

Trigonometry helps us analyze angles and sides of triangles, which are essential in vector decomposition. In physical scenarios like navigating with a treasure map, trigonometry allows us to break down complex paths into simpler components.

When you move in different directions, your movement can be broken down into horizontal and vertical components. This is often done using sine and cosine functions.

• The cosine of an angle helps determine the adjacent side of a right triangle in relation to the hypotenuse. In vector terms, it helps find the component along the axis that's closely aligned with the direction of movement.
• The sine function provides the length of the opposite side in relation to the hypotenuse, helping to determine the component perpendicular to the initial direction.

Applying these principles, for example, if you walk 1.25 km at 30.0° west of north, you use cosine for the north-south component and sine for the east-west component, adjusting signs based on direction. Trigonometry enables precise navigation by transforming angular movement into linear paths, guiding you directly back to your starting place when reversing these components.

Displacement Calculation

Displacement calculations give you the total change in position, not just the distance traveled. Adding vector components allows you to find the starting point relative to the endpoint in the quickest straight line.

In the treasure map problem, we're concerned with calculating the resultant displacement after following a path composed of three segments.

• First, convert all movements into the same units for consistent calculation—here, everything is converted into meters.
• Break down each segment into north-south and east-west components using trigonometric functions.
• Sum these components separately. The result gives the overall shift in position, not just summing distances.

For example, the sum of north-south components includes a negative value for movement directly south and positive calculated values from the angled paths. By calculating both direction components' sums, you obtain a single vector showing the final displacement. This vector tells you how far you need to walk and in which direction to return to the oak tree from the treasure's location.

Graphical Solution

A graphical solution is a visual check of the calculated displacement. This step confirms if the mathematical results align with a practical visualization.

To implement this, you'd draw each segment of the path to scale on graph paper:

• Ensure each angle is drawn accurately using a protractor and the proper distances with a ruler, based on the problem's specific directions.
• Trace your movements, marking starting and ending points clearly.
• Draw the return vector directly from the final position back to the start.

This visual path should align closely with your calculated results if drawn to scale correctly. It acts as a sanity check, making sure the direction and magnitude of the return path are correct. Tools like graph paper and protractors make such visual corrections straightforward and offer an intuitive understanding of how vector components sum up in real-life scenarios.