Module 2 - Motion in 2D

Vectors and Scalars

There are two different types of variables used in this physics class, scalars and vectors.

Scalar: a scalar is a normal number or variable like you have used in math classes. the ’value’ associated with this variable is known as the magnitude. Some examples of scalar quantities are mass, time, temperature, and moment of inertia. Note that scalars can also be unit-less values such as coefficient of friction and phase angle.

Vector: A vector quantity also has a value (referred to as its magnitude), but now includes added information: the direction. There are many vector quantities that you will study in this class. Examples are position velocity acceleration force momentum etc.

Fortunately, you do not have to memorize which type of variable is a vector and which is a scalar. You can simply think about the variable and figure out if it is a vector or a scalar. I will illustrate this with two examples:

When defining the mass of an object (let’s say your shoes), you can completely define the quantity by simply stating the number of kilograms of mass the object has. It does not make sense to say that the shoes have a mass of m=0.5kg in the upward direction. The lack of directional information means that this quantity is a scalar.

Imagine you are trying to describe your movements around your house. If you move from your desk to the bathroom, you will need to give information on which way to move. You cannot simply say “walk a distance of 4.6m and you will arrive in the bathroom”. You may walk the incorrect direction and end up in the kitchen instead. The requirement of this directional information means that this quantity is a vector.

Vector Maths

Effective communication in Engineering and Science requires you master both of these ideas:

Graphical Vectors: Arrows

The visual way to describe a vector uses an arrow. Drawn (or imagined) arrows are useful because they include both direction (angle) and magnitude (length). This description of a vector is often the starting point for solving a physics problem.

Mathematical Vectors (x,y,z)

A picture is very useful for helping get your head around a problem, but it is not useful when actually solving for unknowns in a problem. Instead, you will have to rely on mathematics, which uses “ordered pairs” to describe the vector. As an example, the vector for the position would be written like this: \(\vec{x} = (x,y)\).

Important note This type of mathematics is vast and powerful. You will learn a bunch in this course, and also in Calc3, before needing it classes like Statics. Learn what you can now before you need it in later!

The following video shows the basics of vectors and coordinate systems:

Vector Multiplication

Multiplication with vectors is a bit more complicated than in basic algebra. There are 3 types to discuss for vectors in 3D:

Scalar multiplication: makes the vector larger/smaller, with no change in direction. Simply multiply the the scalar by each component \(c\vec{A} = (cA_x,cA_y,cA_z)\)
Dot Product: (See Module 5: Energy) Find the amount of vector A which is parallel to B. Changes the output to a scalar (no direction).
Cross Product: (See Module 7: Torque) Find the amount of vector A which is perpendicular to B. Outputs a vector perpendicular to both A and B (THIS IS POWERFUL).

Displacement Velocity and Acceleration in 2D

The vector representation of position, velocity and acceleration require the use of some sophisticated mathematical notation that you will need to get used to. We will continue using this notation for the rest of the course, so practicing it now will be beneficial for later use.

I will start this discussion by motivating that the Displacement, Velocity, and Acceleration are each vectors, and are each allowed to point in different directions. These terms are related to each other, but the relationship has to do with the rate of change, not the actual direction or magnitude.