Motion in Two Dimensions

Definition of Motion in Two Dimensions

Although we observe things moving daily, rarely do we dwell on the question "how many dimensions is this motion occurring in?" One of the simplest examples of movement is an object moving linearly in one dimension. Let's go back to the highway example. A car moving along the x-axis represents motion in one dimension. If that same car was moving in one direction with a constant velocity while accelerating in another direction, it would be two-dimensional motion.

Many of the moving objects we observe and experience every day do not travel in straight lines, and their motion is best described as two-dimensional. In one dimension, we analyze the motion of objects in a straight line by studying displacement, velocity, and acceleration. Now, how can we extend what we have learned about motion in one dimension to two-dimensional situations?

Forces and Motions in Two Dimensions

To make a distinction between force and motion, let's define exactly what we mean by force.

Forces exerted on objects are always due to the interaction of that object with another object, like a push or a pull. An example of a force is throwing a baseball, as the process can be simplified into an object being pushed through the air. Similarly, when friction acts between two surfaces, we must consider both, the frictional force that is parallel to the surface and the normal force perpendicular to it. These are best described using x and y co-ordinates in two dimensions.

Two great examples of two-dimensional motion are projectile and circular, where the relevant forces are gravitational and centripetal, respectively.

Projectile Motion

Projectile motion is a special case of two-dimensional motion.

When analyzing projectile motion, it is critical to remember that it is always best to break the motion up into two components: one along the horizontal axis, and one along the vertical axis.

When a projectile is launched at an angle, the object is moving in both the x and y directions simultaneously. These can be expressed using vectors, where the addition will result in a net vector. Most of the time, projectile motion is modeled without including the effects of air resistance; hence it can be ignored.

When neglecting air resistance, the horizontal velocity component of a projectile has no net force acting upon it, while the vertical component is affected only by the force of gravity, meaning the acceleration term \(a_y\) can be replaced with the acceleration due to gravity \(g\).

Uniform Circular Motion

Another common example of two-dimensional motion is uniform circular motion.

Even though the speed is constant, the velocity is not. This is because the direction of the velocity vector changes from point to point along the path it travels. The fact that the velocity changes means that the acceleration is non-zero.

In circular motion, there is always a centripetal acceleration whose magnitude can be expressed as \(a_\mathrm{c}\),

\[a_\mathrm{c}=\frac{v^2}{r},\]

where \(r\) is the radius of the circle. As a result, there is a centripetal force that points to the center of the circle defined by the object's motion. This force changes the direction of the velocity vector of the object, but not its magnitude. Finally, the time it takes to complete a full revolution is known as a period \(T\) and is given by the equation

\[T=\frac{2 \pi r}{v}.\]

A few good examples of circular motion are a Ferris wheel, or a satellite orbiting a planet.

Formulas for Motion in Two Dimensions

Before we list all the relevant motion equations for two dimensions, let's recall the main equations of motion in one dimension.

Table 1 - Kinematic equations for motion in one dimension with constant acceleration.

Motion in two dimensions is described by the same equations as the ones compiled in Table 1 above, the only difference is in notation. In two-dimensional motion we consider two directions (x and y) so the notation for position becomes:

\[ \Delta \vec{r}=\vec{r}_2-\vec{r}_1,\]

where \( \vec{r}_1\) and \( \vec{r}_2\) are vectors from the origin to the points \( A\) and \( B\) in Figure 2. Each of these vectors has x and y components.

Fig. 2 - Example of motion in two dimensions.

In this case, the average velocity is the displacement vector \( \overrightarrow{A B}\) divided by the time it takes for the object to move from \( A\) to \( B\). Instantaneous velocity, on the other hand, can be expressed as

\[\vec{v}= \frac{\mathrm{d}\vec{r}}{\mathrm{d}t},\]

where the x component is equal to

\[v_x= \frac{\mathrm{d}x}{\mathrm{d}t}.\]

The instantaneous acceleration is the change in velocity over time:

\[\vec{a}= \frac{\mathrm{d}\vec{v}}{\mathrm{d}t}.\]

Since it is not convenient to do calculations with vectors, to solve problems in two-dimensional motion, we use the components of the vectors. Then the vector equations turn into algebraic equations, the same as the equations used in one-dimensional motion, which are easier to solve using elementary mathematical operations. Following that logic, all the equations in Table 1 can be split into x and y components:

\begin{align} v_x&=v_{0 x}+a_x t \\ v_y&=v_{0 y}+a_y t \end{align}

when calculating velocity,

\begin{align} x-x_0&=v_{0 x} t+\frac{1}{2} a_x t^2 \\ y-y_0&=v_{0 y} t+\frac{1}{2} a_y t^2, \end{align}

for obtaining the position components, and

\begin{align} v_x^2&=v_{0 x}^2 +2 a_x(x-x_0) \\ v_y^2&=v_{0 y}^2 +2 a_y(y-y_0). \end{align}

Breaking up expressions into components turns a two-dimensional motion problem into a system of two equations with two unknowns.

It's easier to understand these equations through application in example problems.

Examples of Motion in Two Dimensions

Let's look at an example problem of projectile motion in two dimensions.

Motion in Three Dimensions

Just like converting from one to two-dimensional motion, objects in three dimensions can be analyzed using one-dimensional kinematic relationships if the motion is separated into components. To achieve that, a third axis is introduced, therefore now everything is described using x, y, and z coordinates.

Unit vector notation is applied to represent vectors as the sum of their constituent components in each direction. The usual coordinates of x, y, and z, are represented by \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\), respectively. For instance, a vector \(\vec{r}\) in a three dimension space can be written as

\begin{align} \vec{r}&=(A, B, C) \\ \vec{r}&=A\hat{i}+B\hat{j}+C\hat{k}. \end{align}

Fig. 4 - A gyroscope clearly visualizes motion in three dimensions.

An example of three-dimensional motion is the motion of a gyroscope. More on how to qualitatively describe the motion of a particle in three-dimensional space will be covered in later Mechanics topics such as Electricity and Magnetism.