Uniformly Accelerated Motion

Uniformly Accelerated Motion Definition

Throughout our introduction to kinematics so far, we’ve encountered several new variables and equations to solve problems for motion in one dimension. We’ve paid close attention to displacement and velocity, as well as changes to these quantities, and how different initial conditions affect the overall motion and outcome of a system. But what about acceleration?

Observing and understanding the acceleration of moving objects is just as important in our initial study of mechanics. You may have picked up that so far we’ve primarily been examining systems where acceleration is zero, as well as systems where the acceleration remains constant during some period of time. We call this uniformly accelerated motion.

The attractive force of gravity results in the uniformly accelerated fall of a skydiver, Creative Commons CC0

In other words, the velocity of a moving object uniformly changes with time and the acceleration remains a constant value. Acceleration due to gravity, as seen in the fall of a skydiver, an apple from a tree, or a dropped phone to the floor, is one of the most common forms of uniform acceleration that we observe in our everyday lives. Mathematically, we can express uniform acceleration as:

\begin{align*}a=\mathrm{const.}\end{align*}

Calculus Definition of Acceleration

Recall that we can calculate the acceleration \(a\) of a moving object if we know starting and ending values for both the velocity and time:

\begin{align*}a_{avg}=\frac{\Delta v}{\Delta t}=\frac{v_1-v_0}{t_1-t_0}\end{align*}

where \(\Delta v\) is the change in velocity and \(\Delta t\) is the change in time. However, this equation gives us the average acceleration over the time period. If we want to determine the instantaneous acceleration instead, we need to remember the calculus definition of acceleration:

\begin{align*}a_{inst}=\frac{\mathrm{d}v}{\mathrm{d}t}=\frac{\mathrm{d}^2x}{\mathrm{d}t^2}\end{align*}

That is, acceleration is mathematically defined as the first derivative of the velocity and the second derivative of position, both with respect to time.

Uniformly Accelerated Motion Formulas

It turns out you already know the formulas for uniformly accelerated motion — these are the kinematics equations we learned for motion in one dimension! When we introduced the core kinematics equations, we assumed that all these formulas accurately describe the motion of an object moving one-dimensionally as long as the acceleration is held constant. Before, this was largely an aspect that we implied and didn’t dig further into.

Let’s rearrange our kinematics equations and isolate the acceleration variable. This way, we can easily use any of our formulas to solve for the value of acceleration, given different initial conditions to start. We’ll start with the formula \(v=v_0+at\).

Our next kinematic equation is \(\Delta x=v_0t+\frac{1}{2}at^2\).

Our final kinematic equation of interest is \(v^2=v_0^2+2a \Delta x\).

You may recall that there is an acceleration independent equation associated with kinematics, but this equation is irrelevant here since the acceleration variable is not included.

Uniform Motion vs. Uniform Acceleration

Uniform motion, uniform acceleration — is there really a difference between the two? The answer, perhaps surprisingly, is yes! Let’s clarify what we mean by uniform motion.

Although the definitions of uniform motion and uniformly accelerated movement sound similar, there’s a subtle difference here! Recall that for an object moving with a constant velocity, the acceleration must be zero according to the definition of velocity. Therefore, uniform motion does not also imply uniform acceleration, since the acceleration is zero. On the other hand, uniformly accelerated motion means the velocity is not constant but the acceleration itself is.

Graphs for Uniformly Accelerated Motion

We previously looked at a few graphs for motion in one dimension — now, let’s return to uniformly accelerated motion graphs in a bit more detail.

Uniform Motion

We just discussed the difference between uniform motion and uniformly accelerated motion. Here, we have a set of three graphs that visualize three different kinematics variables for an object undergoing uniform motion during some time frame \(\Delta t\):

We can visualize uniform motion with three graphs: displacement, velocity, and acceleration, MikeRun via Wikimedia Commons CC BY-SA 4.0

In the first graph, we observe that the displacement, or change in position from the starting point, linearly increases with time. That motion has a constant velocity throughout time. The velocity curve in the second graph has a slope of zero, held constant to the value of \(v\) at \(t_0\). As for acceleration, this value remains zero throughout the same time period, as we’d expect.

Another important aspect to note is that the area under the velocity-time graph equals the displacement. Take the shaded rectangle in the velocity-time graph above as an example. We can quickly calculate the area under the curve by following the formula for the area of a rectangle, \(a=b \cdot h\). Of course, you can also integrate to find the area under the curve:

\begin{align*}\Delta s = \int_{t_1}^{t_2} v(t)\,\mathrm{d}t\end{align*}

In words, we can integrate the velocity function between a lower and upper limit of time to find the change in displacement that occurred during that time period.

Uniform Acceleration

We can graph the same three types of plots to examine uniformly accelerated motion. Let’s look at a velocity-time graph:

Linearly increasing velocity with time following the velocity function v(t)=2t, with the area under the curve equaling the displacement, StudySmarter Originals

Here, we have a simple velocity function \(v(t)=2t\), plotted from \(t_0=0\,\mathrm{s}\) to \(t_1=5\,\mathrm{s}\). Since the change in velocity is nonzero, we know the acceleration will be nonzero as well. Before we take a look at the acceleration plot, let’s calculate the acceleration ourselves. Given \(v_0=0\, \mathrm{\frac{m}{s}}\), \(v_1=10\, \mathrm{\frac{m}{s}}\), and \(\Delta t=6\, \mathrm{s}\):

\begin{align*} a=\frac{v_1-v_0}{t} \\ a=\mathrm{\frac{10\, \frac{m}{s} - 0\, \frac{m}{s}} {5\, s}} \\ a=\mathrm{2\,\frac{m}{s^2}} \end{align*}

Now, let’s take a look at the acceleration-time graph:

Acceleration-time graphs for uniformly accelerated motion have a slope of zero. The area under this curve is equal to the change in velocity during the time frame, StudySmarter Originals

This time, the acceleration-time plot shows a constant, nonzero acceleration value of \(2\,\mathrm{\frac{m}{s}}\). You may have noticed here that the area under the acceleration-time curve is equal to the change in velocity. We can double-check this is true with a quick integral:

\begin{align*} \Delta v = \int_{0}^{5}2\,\mathrm{d}t = 2t \\ \Delta v = 2(5)-2(0) \\ \Delta v = 10\, \mathrm{\frac{m}{s}} \end{align*}

Finally, we can continue working backward to calculate the change in displacement in meters, even though we don’t have a graph for this variable in front of us. Recall the following relationship between displacement, velocity, and acceleration:

\begin{align*} \Delta s = \int v(t)\,\mathrm{d}t = \iint a(t)\,\mathrm{d}t \end{align*}

Although we know functions for both velocity and acceleration, integrating the velocity function is easiest here:

\begin{align*}\Delta s = \int_{0}^{5} 2t\,\mathrm{d}t = \frac{2t^2}{2} = t^2 \\ \Delta s = (5)^2 - (0)^2 \\ \Delta s = 25\, \mathrm{m} \end{align*}

Remember that this calculation gives us the net displacement over the five-second time period as opposed to a general function of displacement. Graphs can tell us quite a lot about an object in motion, especially if we’re given minimal information at the start of a problem!

Examples of Uniformly Accelerated Motion

Now that we’re familiar with the definition and formulas for uniformly accelerated motion, let’s walk through an example problem.

Before we wrap up our discussion, let’s walk through one more uniformly accelerated motion example, this time applying the kinematics equations we reviewed earlier.

Uniformly accelerated motion is a crucial component of our early studies in kinematics and mechanics, the physics of motion that governs much of our everyday experiences. Knowing how to recognize uniform acceleration as well as how to approach these problems is an early step toward bettering your understanding of the universe as a whole!