Problem 20 - Fundamentals of Physics

Question

An explorer is caught in a whiteout (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while returning to base camp. He was supposed to travel due north for \(5.6 \mathrm{~km}\), but when the snow clears, he discovers that he actually traveled \(7.8 \mathrm{~km}\) at \(50^{\circ}\) north of due east. (a) How far and (b) in what direction must he now travel to reach base camp?

Step-by-Step Solution

Answer

The explorer must travel approximately 5.023 km at an angle of 4.29° north of west to reach base camp.

Step 1: Understanding the problem

The explorer initially needed to travel due north for \(5.6\, \mathrm{km}\), but instead he traveled \(7.8\, \mathrm{km}\) at \(50^{\circ}\) north of east. We need to determine his current position relative to the starting point and how he can reach the correct destination (base camp) from his current location.

Step 2: Setting up the coordinate system

We place the starting point of the explorer at the origin \((0, 0)\) of a coordinate system with the positive \(y\)-axis pointing north and the positive \(x\)-axis pointing east. The base camp's position is \((0, 5.6)\) since it is \(5.6\, \mathrm{km}\) directly north.

Step 3: Calculating the explorer's misaligned position

To find the explorer's position after traveling \(7.8\, \mathrm{km}\) at \(50^{\circ}\) north of east, we decompose this into horizontal and vertical components. The formulas are:- \(x = 7.8 \cdot \cos(50^{\circ})\)- \(y = 7.8 \cdot \sin(50^{\circ})\)Substitute the values to find:\(x \approx 7.8 \cdot 0.643 = 5.009\)\(y \approx 7.8 \cdot 0.766 = 5.975\)The explorer's position is \((5.009, 5.975)\).

Step 4: Calculating the displacement needed to reach the base camp

We calculate the vector needed to travel from the explorer's current position to the base camp's position. The components of this vector are:- \(x_{basecamp} - x_{current} = 0 - 5.009 = -5.009\)- \(y_{basecamp} - y_{current} = 5.6 - 5.975 = -0.375\).

Step 5: Finding the magnitude (distance) of the required travel

The magnitude of the vector gives the distance the explorer needs to travel to reach the base camp. This is calculated using the Pythagorean theorem:\[ d = \sqrt{(-5.009)^2 + (-0.375)^2} \approx \sqrt{25.09008 + 0.140625} \approx \sqrt{25.230705} \approx 5.023 \text{ km} \]

Step 6: Calculating the direction (angle) of the required travel

To find the direction, we calculate the angle \(\theta\) with respect to the positive east direction (the positive \(x\)-axis) using:\[ \theta = \tan^{-1}\left( \frac{\Delta y}{\Delta x} \right) = \tan^{-1}\left( \frac{-0.375}{-5.009} \right) \]This simplifies to:\[ \theta \approx \tan^{-1}(0.0749) \approx 4.29^{\circ} \]Since both components are negative, this suggests the direction is \(4.29^{\circ}\) north of west.

Key Concepts

Coordinate System

A coordinate system is an essential tool in mathematics and physics, allowing us to visually and conceptually understand spatial relationships. Imagine it as a grid overlaying a plane, with two main axes: the x-axis typically running horizontally, and the y-axis running vertically. Each point on this plane can be described in terms of these axes using coordinates.

In the problem, we use a specific coordinate system to map the explorer's journey. The starting point is set at the origin, denoted as (0, 0). Here, the positive y-axis points north, while the positive x-axis points east. This setup allows easy translation of directions into mathematical terms.

• North corresponds to movement along the positive y-axis.
• East corresponds to movement along the positive x-axis.

By using this coordinate system, we can clearly define the destination as (0, 5.6), meaning 5.6 km due north of the starting point. This coordinate-based framework helps in breaking down the complex movement of the explorer into understandable components that can be analyzed separately.

Angle Calculation

Calculating angles is a critical part of understanding direction in a coordinate system. In our problem, this involves evaluating the direction of the explorer's path using degrees.

The explorer intended to head due north but veered off to travel at an angle of 50 degrees north of east. Angles are typically measured from a reference line; in this case, due east is our reference. This angle means the path forms a 50-degree angle upwards from the horizontal x-axis.

We need to determine the angle he must head to reach his intended destination after veering off. We use the arctangent (\( \tan^{-1} \) function to calculate this angle. The formula for angle calculation with displacement components is:

• \( \theta = \tan^{-1} \left(\frac{\Delta y}{\Delta x}\right) \)

After solving, we find the necessary angle to adjust his course, providing insight into the precise direction needed, taking into account both the position and distance traveled.

Trigonometry

Trigonometry offers us the tools to work with angles and distances in the study of triangles. When dealing with directions and distances, it becomes invaluable, allowing us to break vectors into components that are easier to analyze separately.

The explorer's path can be divided into a hypotenuse, opposite, and adjacent sides of a right triangle using trigonometric functions:

• The cosine function relates the adjacent side (x-component) to the hypotenuse (the path length). The formula is: \( x = r \cdot \cos(\theta) \).
• The sine function relates the opposite side (y-component) to the hypotenuse. The formula is: \( y = r \cdot \sin(\theta) \).

Applying these formulas helps us find where the explorer ended up after traveling off course. Trigonometry thus helps bridge the gap between abstract angles and real-world distances in the coordinate system.

Displacement

Displacement refers to the change in position of an object. It's a vector quantity, which means it has both magnitude and direction, distinguishing it from mere distance, which is scalar.

In the explorer's journey, displacement was the difference between his intended final point and his current position after traveling off course. To find this:

• Calculate the difference in x and y coordinates between current and destination positions.
• Use the Pythagorean theorem to find the magnitude of this displacement vector: \[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \]

This solution not only gives the direct path length to the camp but also tells us in which direction to orient his travel. Therefore, understanding displacement is crucial to guide the explorer correctly back to base camp.