Question

A car is driven east for a distance of $54\mathrm{km}$, then north for $32\mathrm{km}$, and then in a direction ${28}^{\circ }$ east of north for $27\mathrm{km}$. Draw the vector diagram and determine the total displacement of the car from its starting point.

Short answer

The total displacement of the car from its starting point can be calculated by using the vectors representing each part of the journey, in conjunction with the laws of sines and cosines to consider both the distance and direction of each vector. This involves quite a bit of calculation and breaking the problem down into solvable parts.

Step-by-step solution

Visualizing the Displacements as Vectors

Start by drawing an eastward arrow to represent the car's first movement. Label this vector as 54 km. Next, from the end point of this vector, draw a vector pointing north representing the car's second movement. Label this as 32 km. To depict the car's final movement, draw from the end point of the second vector, a vector which is ${28}^{\circ }$ east from the north. Label it as 27 km

Using the Laws of Cosines and Sines to Combine the Vectors

We want to determine the total displacement of the car, which would be represented by a vector from the starting point to the last point. However, we can't simply add the three distances together, because they're in different directions. We can solve this by breaking the problem down into two dimensions: east-west and north-south. We can solve the lengths of the sides formed by the 32km and 27 km vectors (label this as a) using the Law of Cosines: $a^2={32}^2+{27}^2-2\ast 32\ast 27\ast cos(28)$ and the angle between the 54 km and this new total vector (label this as θ) using the Law of Sines: $sin(\theta )=27\ast sin(28)/a$

Calculating Final Displacement

The total displacement is a vector from the starting point to the end point. Its eastward part is measured by adding the eastward displacement (the distance that the car traveled east) and the eastward component of the final part of the journey: $54+a\ast cos(\theta )$. Its northward part includes the northward displacement and the northward component of the final part of the journey: $32+a\ast sin(\theta )$. The magnitude of total displacement is calculated by Pythagoras theorem $sqrt(54+a\ast cos(\theta ){)}^2+(32+a\ast sin(\theta ){)}^2$, and the direction is calculated by $atan[(32+a\ast sin(\theta ))/(54+a\ast cos(\theta ))]$

Key concepts

Vector Addition

Vector addition is a fundamental concept in physics and mathematics, where two or more vectors are combined to find a resultant vector. In our context, this concept applies when we're calculating the total displacement of an object, like the car in our exercise, which has moved in different directions.

Imagine you're walking your dog: you take a few steps forward, then to the side, and then diagonally. Your 'journey' can be broken down into separate legs, each represented by a vector that has both a magnitude (how far you went) and a direction (which way you went). To find out the total distance and the final direction from the point you started, you need to 'add' these vectors following specific rules.

The most basic form of vector addition is when two vectors are in the same line. In this case, you simply add or subtract their magnitudes depending on their direction. But it gets a touch trickier if the vectors are not aligned, as you have to take into account both magnitude and direction. You cannot just add up distances because it would disregard directions. Instead, use vector components or geometrical methods, both of which are precise approaches to solving vector problems.

Law of Cosines

The Law of Cosines is an essential tool for finding unknown parts of a triangle when you don't have a right angle. It's the extended rule to the Pythagorean theorem that works for any type of triangle.

The formula states that in any triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, minus twice the product of those sides and the cosine of the included angle. Mathematically, for a triangle with sides of lengths a, b and c, where θ is the angle opposite side c, it goes: $c^2=a^2+b^2-2ab\cos (\theta )$.

Using this in our exercise allows us to calculate the unknown side of the non-right angle triangle formed by the car's movements without needing a right angle. This lets us find one side of the triangle, which is also a part of the total displacement vector.

Law of Sines

The Law of Sines is a powerful relationship between the lengths of the sides of a triangle and the sines of its angles. It's the go-to tool when we know either two angles and one side or two sides and one angle of a triangle, but it's not a right triangle.

The formula states: $\frac{a}{\sin (\alpha )}=\frac{b}{\sin (\beta )}=\frac{c}{\sin (\gamma )}$, where a, b, and c are the triangle's sides, and α, β, and γ are the respective opposite angles. When we apply the Law of Sines to our car's trajectory problem, we can find the direction of the overall displacement, since we have used the Law of Cosines to find the lengths of the sides. Knowing one angle from the car's final turn and the side opposite to it allows us to find the other needed angles and thus the orientation of the displacement vector.

Vector Diagram

A vector diagram is like a map that uses arrows to represent vectors. It shows the magnitude and direction of each vector and is particularly useful for visualizing problems in physics involving forces or displacements.

To create a vector diagram, each vector is drawn to scale, with an arrowhead pointing in the vector's direction. When vectors are added, they are placed tip-to-tail, meaning the start of one vector begins at the tip of the previous one. This step-by-step construction allows us to see the total effect of all vectors combined, which in the case of our car problem, results in a clear visual idea of how the total displacement vector is formed from the individual movements.

By drawing the paths taken by the car as vectors in our exercise, we can use the diagram to explore the problem geometrically and calculate the final displacement vector, which represents the car's overall change in position from the starting point.