Problem 60 - College Physics

Question

To go to a football stadium from your house, you first drive \(1000 \mathrm{~m}\) north, then \(500 \mathrm{~m}\) west, and finally \(1500 \mathrm{~m}\) south. (a) Relative to your home, the football stadium is (1) north of west, (2) south of east, (3) north of east, (4) south of west. (b) What is the straight-line distance from your house to the stadium?

Step-by-Step Solution

Answer

(a) South of west, (b) 707.11 m

Step 1: Determine Displacement in North-South Direction

First, calculate the total displacement in the north-south direction. You drive 1000 m north and then 1500 m south. The north-south displacement is calculated as follows:\[\text{North-South Displacement} = 1500 \text{ m south} - 1000 \text{ m north} = 500 \text{ m south}\]

Step 2: Determine Displacement in East-West Direction

Next, calculate the displacement in the east-west direction. You drive 500 m west, so the east-west displacement is:\[\text{East-West Displacement} = 500 \text{ m west}\]

Step 3: Determine Overall Direction Relative to Home

Using the displacements computed, the total displacement relative to your home is 500 m south and 500 m west. This places the stadium in the south of west relative to your starting point.

Step 4: Calculate the Straight-Line Distance

Use the Pythagorean theorem to determine the straight-line distance from your house to the stadium. The formula for the distance based on the right triangle formed by the north-south and east-west displacements is:\[\text{Distance} = \sqrt{(500 \text{ m})^2 + (500 \text{ m})^2} = \sqrt{250000 \text{ m}^2 + 250000 \text{ m}^2} = \sqrt{500000 \text{ m}^2} = 707.11 \text{ m}\]

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in geometry, particularly useful in calculating distances in right-angle triangles. This theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it can be expressed as:\[ c^2 = a^2 + b^2 \]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. In the case of the football stadium problem, the north-south and east-west displacements form the two smaller sides of a right triangle. By applying the Pythagorean theorem, we can easily calculate the straight-line distance from the house to the stadium, effectively giving us the length of the hypotenuse. This principle is incredibly useful whenever you need to find direct distances in two-dimensional space, such as in navigation, architecture, or physics. Remember, applications of this theorem are endless, from finding the distance between two cities on a map to computing the length of a diagonal in a room.

north-south displacement

North-south displacement refers to the movement along the north and south directions. It's a component of directional movement that helps in determining the overall path taken or needed. In the problem, the initial part of the journey involves moving 1000 meters northward. Afterward, there is a displacement of 1500 meters towards the south.
The north-south displacement is calculated by subtracting the movement towards the north from the movement towards the south:\[\text{North-South Displacement} = 1500 \text{ m south} - 1000 \text{ m north} = 500 \text{ m south} \]This value indicates that after completing the journey, there is a net movement of 500 meters towards the south. Displacement is different from total distance; it's a vector quantity indicating both magnitude and direction. It's crucial to understand this when solving problems involving multiple movements or directions, as displacement highlights how far off and in what direction you end up from your original position.

east-west displacement

East-west displacement deals with the horizontal movement across east and west directions. In many problems, including the football stadium scenario, it represents part of the path taken through these two cardinal directions. Here, there is a straightforward journey of 500 meters heading west.
Unlike the north-south displacement, there's no return displacement in the east direction in this problem, making:\[\text{East-West Displacement} = 500 \text{ m west} \]This direct movement leads to a straightforward calculation. Like the north-south displacement, east-west displacement is also a form of vector, indicating both direction and distance. Each of these displacements helps to form a right triangle through which the straight-line distance to the destination, in this case, the stadium, can be accurately calculated using geometrical methods. Understanding displacements separately allows us to efficiently summarize the total movement's direction and magnitude, which is essential for precise navigation or solving physics problems.