Problem 85 - Fundamentals of Physics

Question

You are kidnapped by political-science majors (who are upset because you told them political science is not a real science). Although blindfolded, you can tell the speed of their car (by the whine of the engine), the time of travel (by mentally counting off seconds), and the direction of travel (by turns along the rectangular street system). From these clues, you know that you are taken along the following course: \(50 \mathrm{~km} / \mathrm{h}\) for \(2.0 \mathrm{~min}\), turn \(90^{\circ}\) to the right, \(20 \mathrm{~km} / \mathrm{h}\) for \(4.0 \mathrm{~min}\), turn \(90^{\circ}\) to the right, \(20 \mathrm{~km} / \mathrm{h}\) for \(60 \mathrm{~s}\), turn \(90^{\circ}\) to the left, \(50 \mathrm{~km} / \mathrm{h}\) for \(60 \mathrm{~s}\), turn \(90^{\circ}\) to the right, \(20 \mathrm{~km} / \mathrm{h}\) for \(2.0 \mathrm{~min}\), turn \(90^{\circ}\) to the left, \(50 \mathrm{~km} / \mathrm{h}\) for \(30 \mathrm{~s}\). At that point, (a) how far are you from your starting point, and (b) in what direction relative to your initial direction of travel are you?

Step-by-Step Solution

Answer

a) 0.67 km from start; b) ~6.8° left of initial direction.

Step 1: Understanding the problem

To determine your final position and direction, you need to follow the path based on the given speed, time, and turns. This involves calculating distances, considering turns and calculating the resulting displacement from the starting point.

Step 2: Convert time to hours

For each segment of travel, convert the given time from minutes or seconds to hours by dividing minutes by 60 and seconds by 3600. This is necessary because speed is given in kilometers per hour.

Step 3: Calculate distances traveled

Multiply the speed by the time (in hours) for each segment to find the distance traveled. For instance, for the first segment: \[ \text{Distance} = 50 \text{ km/h} \times \frac{2}{60} \text{ h} = 1.67 \text{ km} \]

Step 4: Track position changes with directional turns

Start from the origin (0,0). For each segment, adjust your position on a coordinate grid relative to direction changes and the calculated distances. Assume starting direction along the positive x-axis.

Step 5: Plot each segment and update position

1. First segment: Go 1.67 km along +x. 2. Right turn: Go 1.33 km along -y. 3. Right turn: Go 0.33 km along -x. 4. Left turn: Go 0.83 km along +y. 5. Right turn: Go 0.67 km along -x. 6. Left turn: Go 0.42 km along +y.

Step 6: Use Pythagorean theorem to find distance from the starting point

Calculate the net displacement in x and y direction from the starting point and use these to find the distance:\[\text{Net } x = (1.67 - 0.33 - 0.67) \text{ km} \approx 0.67 \text{ km} \\text{Net } y = (-1.33 + 0.83 + 0.42) \text{ km} \approx -0.08 \text{ km} \\text{Distance} = \sqrt{(0.67)^2 + (-0.08)^2} \approx 0.67 \text{ km}\]

Step 7: Determine final direction

Calculate the angle using the arc tangent of the ratio of y-displacement to x-displacement to find the direction relative to the original direction:\[\theta = \tan^{-1} \left(\frac{-0.08}{0.67}\right) \approx -6.8^\circ\]This indicates the direction is approximately 6.8 degrees to the left of the x-axis.

Key Concepts

Coordinate System

A coordinate system acts as a reference framework that helps us describe the precise location of a point in a given space. In this exercise, when tracking movement along the streets, we create a coordinate plane where:

• The starting point is typically designated as the origin ( (0,0) ) .
• The positive x-axis can be thought of as your initial direction or path.
• Turns which are 90 degrees to the right or left shift your path to align with the y-axis either in positive or negative direction.

Whenever there is motion, updating positions within this grid becomes crucial for calculating displacement effectively. In practical terms, the coordinate system assists in keeping track of how each movement and turn translates into coordinate changes, allowing you to figure out where you end up relative to where you began. Thus, it's a central concept for understanding spatial relationships in vector problems like this one.
Including such a logical structure is critical since it provides clarity in visualizing shifts in direction and distance, making problem-solving more straightforward.

Vector Mathematics

Vector mathematics is essential for calculating physical quantities along defined directions. In this displacement problem, each segment of movement forms a vector with two main characteristics: magnitude and direction:

• Magnitude: This is the length of the vector, which represents the distance the car travels in each segment. You compute it by multiplying the speed by the time (both converted to consistent units such as kilometers per hour and hours).
• Direction: Vectors have directional properties, which change with each 90-degree turn. These direction changes can be graphically envisioned as rotating the vector 90 degrees around the coordinate plane.

Tracking the journey along different vectors and updating each on the coordinate grid through their horizontal and vertical components is what gives the resultant path. Adding or subtracting these vectors (based on their respective directions) computes the overall journey's net vector.
This makes vector mathematics extremely powerful for solving multidimensional problems where direction changes occur.

Pythagorean Theorem

The Pythagorean theorem is employed to compute the straight-line distance from the starting point to the final location, even when the path taken is convoluted. Once the net changes in the x and y directions are known, apply the theorem as follows:

• Consider the net x-distance and y-distance as the two shorter sides (legs) of a right triangle.
• The hypotenuse of this triangle represents the direct line distance (or displacement) from the start to the finish.
• Calculate it using the formula: \[\text{Distance} = \sqrt{(\text{Net } x)^2 + (\text{Net } y)^2}\]

For example, in the exercise, using the calculated values of approximately 0.67 km for the net x and -0.08 km for the net y, the computed displacement or the straight-line path distance turned out to be approximately 0.67 km. This method efficiently provides a real understanding of how far you've moved regardless of the route's complexity.