Displacement, Time and Average Velocity

Defining Displacement in Physics

Perhaps the most easily observable aspect of motion is differences in position through time. Whenever you get up from your chair and walk to the kitchen, take the bus from home to school, or walk from your front door to the mailbox, you change your body’s physical position in space. We understand this concept of a positional change to be displacement.

There are several ways we can measure displacement. We can use a standard Cartesian coordinate plane and calculate the difference between various points. Alternatively, we can calculate positional changes between different buildings, towns, states, or countries relative to one another on a map. We can even use GPS coordinates. Regardless of how you decide to measure, you’ll need to remember to define the origin, positive directions, and negative directions first — without knowing these details, you might end up with a wrong calculation!

Distance Versus Displacement

You should be pretty familiar with the concept of distance from traveling in day-to-day life. You probably know how many miles it takes to get to the grocery store from your house, a friend or extended family member’s home, or another destination you often visit. Now, let’s define what we mean by distance in a physics context.

Now distance is sometimes confused with the term distance traveled.

So, what’s the difference between all these terms? Unlike displacement, which can have a negative, positive, or zero value, distance measurements are always non-negative. If this sounds a bit confusing, let’s perform a quick thought experiment. Imagine you are a track runner competing in the \( 400\,\mathrm{m} \) race. The gun sounds and you begin to make your way around the track toward the finish line. Now the length of your path from start to finish is \( 400\,\mathrm{m}, \) however, the displacement from start to finish is \( 0\,\mathrm{m} \) in every direction.

Why? Well, your overall position did not change as you ended in the same position you started. As a result, the distance between start and finish is also \( 0\,\mathrm{m} \) since distance is the magnitude of displacement and the magnitude of zero is zero.

Fig. 1 - A track race as an example of distance traveled versus distance.

Distance and Displacement Formulas

Now that we’re comfortable with the differences between distance and displacement, let’s look at the formulas we need to know. We write the formula for displacement mathematically as

\begin{align*} \text{displacement} &=\text{final position} - \text{initial position,} \\ \Delta x&=x_\text{f}-x_\text{i}, \end{align*}

where \(x_\text{i}\) is the initial position and \(x_\text{f}\) is the final position. The Greek letter \(\Delta\), pronounced “Delta”, indicates a change in some variable. Thus, by \(\Delta x\) we mean a change in position or displacement.

If you know the length of each leg of a straight-line trip between several pairs of points, you can calculate the distance by simply finding the sum of all the individual lengths. We can also calculate the distance \(d\) between two points in a two-dimensional plane using the formula:

\begin{align*} {d =\sqrt{(x_\text{f}-x_\text{i})^2+(y_\text{f}-y_\text{i})^2.}} \end{align*}

In words, we’re calculating the magnitude of the displacement vector, which results in a scalar quantity. Both distance and displacement are measured in units of length, with a corresponding SI base unit of meters, represented by the symbol \(\mathrm{m}\).

Let’s walk through an example comparing displacement and distance formulas to see how these calculations can end up very different in practice.

In the previous example, your displacement is zero miles because there is no change in your initial and final positions. In other words, there is no change in your position since you began and ended at home. Let’s consider another example, this time with a trip ending at a position different from the origin.

The displacement in the previous example is \(\mathrm{-2\, mi}\) because both the pet store and bakery are located on the positive \(x\)-axis to the right of \(x=0\,\mathrm{mi}\) while the school is located on the negative \(x\)-axis. The school is two miles from your house, but in the opposite direction from the pet store and bakery. Let’s look at one more example comparing distance and displacement calculations.

Let’s recap what we’ve learned about distance and displacement so far.

• Distance is the magnitude of displacement and is always non-negative.
• Distance traveled is the length of the path taken from start to finish.
• Distance does not account for direction and is a scalar quantity.
• Displacement is the change in position between two points and can be zero, positive, or negative.
• Displacement depends on the direction and is a vector quantity.

Time Formula in Physics

The concept of time is already a very familiar part of daily life. You have a school schedule to change classes at specific hours of the day, an alarm set for a certain hour to get up in the morning, and an idea of what portion of a day certain tasks will take up. In physics, time is an important variable for understanding all sorts of physical systems, and is notably observable by a change in some quantity.

We measure time in units of seconds, \(\mathrm{s}\), as it is the SI base unit for time. In a practical sense, such as during a lab, we measure the passage of time with a stopwatch or a regular clock. We can also determine the time passed for a moving object using the formula,

\begin{align} \mathrm{time} &=\frac{\text{distance traveled}}{\text{speed}}, \\t&=\frac{s}{v}. \end{align}

Of course, time is a scalar quantity, mathematically determined using other scalar quantities, and measured relative to a previous timestamp chosen on a clock. We understand time to continuously move forward, with no negative direction or equivalent, and no way to undo what’s already been done in the past. We use time as a measurement of how long an event lasted.

Average Velocity Definition

An object experiencing a change in position has a measurable rate of change known as velocity.

Velocity is another way of saying “The object moves this much distance for each unit of time that passes”. Average velocity is simply the average rate of positional change over an entire time period, as opposed to instantaneous velocity, which is measured at a specific moment in time using a given velocity function.

Speed Versus Velocity

Just as there is a key difference between distance and displacement, the same difference exists for speed and velocity.

Speed describes how fast an object moves through space with respect to time, or how much distance an object covers during some specified time period. In everyday language, we might use the terms velocity and speed interchangeably, but in physics, we make an important distinction between the two. Speed is a scalar quantity, a numerical value without a direction, while velocity is a vector quantity, with both magnitude and direction.

Speed and Average Velocity Formulas

Depending on the system at hand and the initial conditions given, there are several formulas we can use in physics to determine the average velocity and speed. The simplest formula for average velocity is,

\begin{align*} \text{average velocity} &= \frac{\text{displacement}}{\text{elapsed time}}, \\ v_{\mathrm{avg}}&=\frac{\Delta x}{\Delta t}, \\ v_{\mathrm{avg}}&=\frac{x_\text{f} - x_\text{i}}{t_\text{f} - t_\text{i}}. \end{align*}

We can calculate the average speed of a moving object using a similar formula:

\begin{align*}\text{average speed}&= \frac{\text{distance traveled}}{\text{elapsed time}} =\frac{s}{t}.\end{align*}

Both velocity and speed are measured in units of \(\mathrm{\frac{length}{time}}\), with the most common unit being \(\mathrm{\frac{m}{s}}\). Let’s walk through a brief example of calculating the speed of a moving car.

Let’s walk through an example of calculating the average velocity using the equation for average velocity.

To finish our discussion on average velocity, let’s take a brief look at finding velocities from a graph.

Graphing Average Velocity

In addition to numerically solving for the average velocity, it’s also useful to graph different variables of motion to visualize the problem at hand. We can use a position-time graph as a tool for examining the velocity of an object given a position function. Let’s use the following graph to practice finding the velocity between a few different points along a curve.

Figure 4: Graph of position plotted as a function of time, where the slope between any two points is equal to the average velocity.

We can find the average velocity by calculating the slope between two points along the curve. Let’s calculate the average velocity for three different segments along the graph using two points from the graph. We’ll be using our formula \(v_{\mathrm{avg}}=\frac{\Delta x}{\Delta t}\) for each calculation.

First, let’s find the average velocity between the second point \((4,8)\) and the fourth point \((12,2)\):

\begin{align*} v_{\mathrm{avg}}= \mathrm{\frac{2\, m-8\, m}{12\, s-4\, s}=-0.8\, \frac{m}{s}}. \end{align*}

Here, the average velocity is negative, and we can see a downward trend on the graph. Next, let's find the average velocity between the third point \((8,6)\) and the fifth point \((18, 6)\):

\begin{align*} v_{\mathrm{avg}}= \mathrm{\frac{6\, m-6\, m}{18\, s-8\, s}=0\, \frac{m}{s}}. \end{align*}

The average velocity is zero because there is no change in position. Finally, let’s calculate the average velocity between points one \((1, 3)\) and two \((4,8)\):

\begin{align*} v_{\mathrm{avg}}= \mathrm{\frac{8\, m-3\, m}{4\, s-1\, s}=2\, \frac{m}{s}}. \end{align*}

Between points one and two, there is an upward trend, so the average velocity is positive.