Problem 13 - Fundamentals of Physics

Question

A person desires to reach a point that is \(3.40 \mathrm{~km}\) from her present location and in a direction that is \(35.0^{\circ}\) north of east. However, she must travel along streets that are oriented either north-south or east-west. What is the minimum distance she could travel to reach her destination?

Step-by-Step Solution

Answer

The minimum distance is 4.73 km.

Step 1: Understanding the Problem

The person needs to travel to a point that is 3.40 km away at an angle of 35° north of east, but she can only travel in straight lines either north-south or east-west. We need to find the minimum distance when traveling in these directions.

Step 2: Breaking Down the Travel Directions

We will break the traveled path into two components: one eastward and one northward, forming a right triangle. The distance along the east-west street corresponds to the base, and the distance along the north-south street corresponds to the height of the triangle.

Step 3: Calculate the Eastward Component

Use the cosine of the angle to find the eastward component: \( x = 3.40 \times \cos(35.0^\circ) \). This calculation gives the distance traveled in the eastward direction.

Step 4: Calculate the Northward Component

Use the sine of the angle to find the northward component: \( y = 3.40 \times \sin(35.0^\circ) \). This calculation gives the distance traveled in the northward direction.

Step 5: Apply the Values to the Formulas

Calculate the values: \( x = 3.40 \times \cos(35.0^\circ) \approx 2.78 \) km and \( y = 3.40 \times \sin(35.0^\circ) \approx 1.95 \) km.

Step 6: Sum of the Distances

Adding the northward and eastward distances gives the total minimum distance: \(2.78 + 1.95 = 4.73\) km.

Key Concepts

Right Triangle

When dealing with navigation based on specific directions and distances, understanding the concept of a right triangle is fundamental. In our problem, the goal is to reach a destination by only traveling along north-south or east-west streets, the path forms a right triangle. This happens because the direction "north of east" at 35° essentially divides the travel direction into two perpendicular components.

• The "eastward" component forms one leg of the triangle.
• The "northward" component forms the other leg.
• The direct path towards the destination represents the hypotenuse.

Breaking down the journey in this manner allows us to apply trigonometric principles to solve for these components, enabling an accurate calculation of the minimum path distance while adhering to street constraints.

Trigonometric Functions

Trigonometric functions are essential tools in decomposing a vector into its components in problems involving right triangles. In our scenario, these functions allow us to derive the "eastward" and "northward" distances using the given hypotenuse and angle.

Cosine Function

The cosine function is used here to find the eastward component. In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. Thus, the eastward distance is calculated as:\[ x = 3.40 \times \cos(35.0^\circ) \]

Sine Function

The sine function helps determine the northward component. It represents the ratio of the opposite side to the hypotenuse. Consequently, the northward distance can be expressed as:\[ y = 3.40 \times \sin(35.0^\circ) \]These functions are vital in accurately breaking down diagonal travel paths, making it easier to compute total travel distance along each leg of the triangle.

Distance Calculation

Distance calculation, in this context, involves finding the sum of the individual paths taken along the grid of streets. After determining the components using trigonometric calculations, the next step is to add these distances together.Given:

• Eastward distance (x) is calculated as \(2.78\) km.
• Northward distance (y) is calculated as \(1.95\) km.

The total distance is simply the sum of these two components:\[ \text{Total distance} = x + y = 2.78 + 1.95 = 4.73 \text{ km} \]This calculated distance represents the least distance the person needs to travel, respecting the street grid, illustrating how vector decomposition and trigonometry collectively provide practical solutions in navigational tasks.