Problem 1 - Applied Mathematics

Question

The resultant of displacements \(2 \mathrm{~m}\) South, \(4 \mathrm{~m}\) West, \(5 \mathrm{~m}\) North is of magnitude: (a) \(3 \mathrm{~m}\) (b) \(7 \mathrm{~m}\) (c) \(5 \mathrm{~m}\) (d) \(\sqrt{65} \mathrm{~m}\) (e) \(11 \mathrm{~m}\).

Step-by-Step Solution

Answer

5 m

Step 1: Break down the displacements

Separate the north-south and east-west components of the displacements. The displacements are: 2 m South, 4 m West, and 5 m North.

Step 2: Find the net north-south displacement

Subtract the southward displacement from the northward displacement. Net north-south displacement = 5 m North - 2 m South = 3 m North.

Step 3: Find the net west-east displacement

Since there is only a westward displacement, the net west-east displacement is 4 m West.

Step 4: Use Pythagorean theorem to find the magnitude of the resultant displacement

The resultant displacement can be found using the Pythagorean theorem: \[ \text{Resultant displacement} = \sqrt{(\text{North-South displacement})^2 + (\text{West-East displacement})^2} \]Substitute the values: \[ \sqrt{(3 \, \text{m})^2 + (4 \, \text{m})^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{m} \]

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a powerful tool for solving problems involving right-angled triangles. Imagine you have a right-angled triangle. According to the theorem, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
The formula is: \[c^2 = a^2 + b^2\] where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. This theorem helps us find unknown lengths and is essential in determining the resultant displacement in this exercise.

Vector addition

To solve problems involving multiple displacements, we use vector addition. Vectors represent quantities that have both magnitude and direction. In this exercise, displacements are vectors. We must add them to find the resultant vector.
Here's how you add vectors:

• Break down each vector into its components (north-south and east-west in our case).
• Add the components separately to get the resultant vector.

In our exercise, we separated the displacements into north-south and east-west components before adding them.

Magnitude of vectors

The magnitude of a vector is its length. From the previous steps, we have the resultant vector’s components. To find the magnitude of the resultant vector, we use the Pythagorean theorem.
The components of the resultant vector are 3 m north and 4 m west. By applying the Pythagorean theorem, we find its magnitude:
\[ \text{Magnitude} = \sqrt{(3 \text{ m})^2 + (4 \text{ m})^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ m} \] This gives us the length of the resultant vector.

Displacement calculations

Displacement calculations involve determining the overall change in position of an object. In our exercise, we calculated the resultant displacement when moving 2 m south, 4 m west, and 5 m north.
The steps include:

• Separating displacements into components.
• Finding the net north-south and east-west displacements.
• Using the Pythagorean theorem to calculate the magnitude.

Following these steps, we found that the magnitude of the resultant displacement is 5 m.