Displacement, Time and Average Velocity

“Are we there yet?” If you’ve ever been a passenger on a dreadfully long road trip for a summer vacation or holiday break, you know just how much every mile matters. Maybe you asked your driver to speed up a bit, knowing that you might arrive at your destination just a bit sooner. And when the answer to that question was No!”, perhaps you studied your planned route, searching for a shorter path to get a few extra minutes of fun.

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    Even the most seemingly simple scenarios of movement, like a boring road trip to a much-less-boring theme park, boil down to a set of fundamental quantities of kinematics. Your journey of understanding how the world works starts with learning how the most basic systems move: why the physical positions in space, direction of travel, rate of motion, and time passed all matter. In this article, we’ll go over the definitions, along with relevant formulas and examples, of displacement, distance, time, and average velocity. Before you know it, you’ll have a much better understanding of all the reasons for the response, “We get there when we get there!”

    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.

    Displacement is the overall change in an object’s position.

    Displacement is a vector quantity: it has direction and magnitude.

    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.

    Distance is the magnitude of displacement.

    Distance is a scalar quantity: it has magnitude but no direction.

    Now distance is sometimes confused with the term distance traveled.

    Distance traveled is the length of the path taken to get from start to finish.

    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.

    Displacement, Time, and Average Velocity, Track, StudySmarterFig. 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.

    Let’s say you need to go to the local pet store for supplies, located five miles away from your home. You begin your trip at home, travel by car to the store, and return home. What is your distance traveled and displacement at the end of the trip?

    Displacement, Time, and Average Velocity, Position Graph for Single Trip, StudySmarterFigure 2: A graph demonstrating the calculations of distance, distance traveled, and displacement for a single trip.

    Let’s start by calculating the distance traveled. In this case, you completed the five-mile drive twice, so the distance traveled is simply

    \begin{align*} s &=\mathrm{5\, mi+5\, mi=10\, mi}. \end{align*}

    The distance you traveled from start to finish of your trip is ten miles. Next, let’s calculate the displacement,

    \begin{align*} \Delta x &= x_\text{f} - x_\text{i}=0\,\mathrm{mi} \end{align*}

    So, despite traveling ten miles, your displacement as well as distance is zero miles because you ended up in the same position you started.

    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.

    Let’s say instead that after traveling to the same pet store as before, you decide to take a detour. This time, you start from home, travel to the pet store, visit a bakery, and then drive to the school. The bakery is three miles away from the pet store, and the school is ten miles away from the bakery. Find the distance traveled, and your displacement at the end of the trip. Calculate your distance to the starting point as well.

    Displacement, Time, and Average Velocity, Position Graph for Multi-leg Trip, StudySmarterFigure 3: A graph demonstrating the calculations of distance, distance traveled, and displacement for a multiple-leg trip.

    Again, let’s sum up the length between each position for the three legs of the trip to find out the distance we traveled:

    \begin{align*} s&=\mathrm{5\, mi+3\, mi+10\, mi=18\, mi}. \end{align*}

    Finally, let’s find the positional difference between the school and home to get our total displacement:

    \begin{align*} \Delta x &=\mathrm{-2\, mi-0\, mi=-2\, mi}. \end{align*}

    We have a negative value for displacement after this trip because the school is located to the left of our home and we chose the positive direction to be to the right. Now to calculate distance, we must take the magnitude of our displacement as follows:

    $$\begin{align}\mathrm{distance}&=|-2\,\mathrm{mi}| =2\,\mathrm{mi}\end{align}.$$

    Our distance at the end of this trip to our starting position is \( 2\,\mathrm{mi} \).

    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.

    In the previous examples, the initial position was your house. If your initial position is school, find the distance and net displacement for a trip beginning at the school, visiting the bakery, and ending at the pet store.

    Again, starting with the distance calculation:

    \begin{align*} d&=\mathrm{10\, mi+3\, mi=13\, mi}.\end{align*}

    And finally, the displacement calculation:

    \begin{align*} \Delta x &=8\, \mathrm{mi}-(-2\,\mathrm{mi})=10\, \mathrm{mi}. \end{align*}

    This time, our displacement from the origin is positive.

    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.

    Time is the measurement of how long an event, or an observable change, takes to occur.

    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 the directional rate of change in position.

    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 is the magnitude of 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.

    Note that distance traveled is denoted by \( s. \)

    Let’s say you’re driving a car and traveled a distance of \( 10.2 \) miles in \( 25 \) minutes. What is your average speed in miles per hour?

    First, we want to convert \(\mathrm{25\, min}\) to \(\mathrm{h}\):

    \begin{align*} \mathrm{\frac{1\, h}{60\, min}\times 25\, min=0.42\, h}. \end{align*}

    Next, we want to use the formula for average speed and solve:

    \begin{align*} \text{average speed}&=\frac{s}{t}, \\ &=\mathrm{\frac{10.2\, mi}{0.42\, h}}, \\ &= 24\, \mathrm{\frac{mi}{h}}. \end{align*}

    Thus, your average speed is \(24\, \mathrm{mi}/h\). Since speed is a scalar quantity, we expected this answer to be non-negative so this is good.

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

    You run \( 100.1\,\mathrm{m} \) to the bus stop, but you drop your notebook at \( 72\,\mathrm{m}. \) Then, you run back to retrieve it in \( 23\,\mathrm{s}. \) Find your average velocity over the 23-second time interval.

    Let’s calculate the velocity using \(x_\text{i}=0\, \mathrm{m}\) and \(x_\text{f}=72\, \mathrm{m}\):

    \begin{align*} v_\mathrm{avg}&=\frac{\Delta x}{\Delta t}, \\ v_\mathrm{avg}&=\mathrm{\frac{72\, m-0\, m}{23\, s}}, \\ v_\mathrm{avg}&=\mathrm{3.1\, \frac{m}{s}}. \end{align*}

    Now, what would your average velocity be if you also dropped your pen at your initial position and run back for it? Say it takes you an additional \( 18\,\mathrm{s} \) to run back to your initial position. Let’s calculate the average velocity of you running to the bus stop and back to retrieve your notebook and pen.

    This time, the total time elapsed is \(\mathrm{23\, s+18\, s=41\, s}\). Now, let’s find your average velocity:

    \begin{align*} v_\mathrm{avg}&=\mathrm{\frac{0\, m-0\, m}{41\, s}} \\ v_\mathrm{avg}&=\mathrm{0\, \frac{m}{s}}. \end{align*}

    Since we only used the endpoints for this calculation, which are both zero, the average velocity is also zero. What is the average speed? Using the formula for average speed, with a total distance traveled of \(\mathrm{200.2\, m}\) from running back and forth between the bus stop, we get

    \begin{align*} \text{average speed}=\mathrm{\frac{200.2\, m}{41\, s}} =4.9\, \mathrm{\frac{m}{s}}. \end{align*}

    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.

    Displacement, Time, and Average Velocity, Position-time Graph, StudySmarterFigure 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.

    Displacement, Time and Average Velocity - Key takeaways

    • Displacement is the overall change in an object’s position.

    • Distance is the magnitude of displacement.

    • Distance traveled is the length of the path taken to get from start to finish.

    • Time is a measurement of how long an event, or an observable change, takes.

    • We calculate the average speed, a measurement of how fast an object moves, with the distance traveled divided by the total time of travel.

    • We calculate the average velocity, a measurement of the direction and rate of travel, with the displacement divided by the time interval spanned.

    • A graphical representation of average velocity is useful for visualizing an object’s path and identifying how the velocity changes over time.

    Frequently Asked Questions about Displacement, Time and Average Velocity

    What is the formula for time in physics?

    The formula for time in physics is displacement divided by average velocity, or t=Δs/Δv.

    What is time in physics?

    Time in physics is the measurement of the duration of an event.

    What does displacement mean in physics?

    Displacement in physics is a vector quantity measuring the change in position with reference to the starting position. Unlike distance, displacement takes the direction of motion into account.

    What is the displacement formula in physics?

    The displacement formula in physics is final position minus initial position, or Δx=x-x₀.

    How do you find the average velocity on a graph?

    The average velocity on a graph is found by calculating the slope between two points along a curve on a position-time graph. The average velocity can also be calculated by finding the area under the curve of an acceleration-time graph.

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