Motion in One Dimension

Motion in One Dimension Formulas

We use the following variables to describe motion in one dimension: position, displacement, distance, speed, velocity, and acceleration. You’ll want to know the meaning of each of these, as well as some important calculus-based definitions for these quantities. Let’s start with the first three variables listed.

Position, Displacement, and Distance

Knowing how to describe where an object is in space will be essential throughout your study of physics. The first variable we use to understand location is position.

Most often, you’ll be using the two-dimensional Cartesian coordinate plane to describe both one and two-dimensional motion. To plot position, both the \(x\) and \(y\) coordinates on a 2D graph represent the object’s location in space.

We use the two-dimensional Cartesian coordinate system to plot and analyze motion in both one and two dimensions, StudySmarter Originals

In the above graph, the initial position of some moving object is \((1,1)\) and the final position is \((3,1)\). The arrow drawn between the starting and ending positions is the displacement of the object.

The \(\Delta x\) is read out loud as “the change in x” or “delta x”. In our example graph above, the displacement is \(\Delta x = (3-1) = 2\) units of length. So, if the units of length were meters, the total displacement of the moving object is \(2\) meters. You might be wondering, how does this relate to the distance traveled — aren’t they the same? The answer is no!

We measure both displacement and distance in units of length, most commonly meters, or \(\mathrm{m}\). The direction of travel is very critical for displacement, so pay attention as you calculate! The displacement of an object can be zero if we end at the same place we started, but the total distance will always be nonzero if we’ve moved at all. In our graph, the distance traveled is the same as the displacement here. However, if the object traveled back to its initial position, then the displacement would be zero with a total distance traveled of \(4\) units.

Speed and Velocity

The next two quantities we use for motion in one dimension are speed and velocity.

Similar to the difference between distance and displacement, the key distinction between speed and velocity is that velocity is a vector quantity, while speed is not.

We measure both speed and velocity in units of length per time, most commonly meters per second, or \(\frac{\mathrm{m}}{\mathrm{s}}\). Differentiating the rate of change in position of a moving object with respect to time will give us the instantaneous velocity, or the velocity measured at a specific moment in time:

$$ v_{x\,(\mathrm{inst})} = \frac{\mathrm{d}x}{\mathrm{d}t}. $$

If instead, we want to find the average velocity, we can use the following formula:

$$ v_{x\,(\mathrm{avg})} = \frac{x_f-x_0}{t_f-t_0} = \frac{\Delta x}{\Delta t}, $$

where \(\Delta x\) is the change in position and \(\Delta t\) is the change in time. This formula is useful if you’re given the numerical values of the starting and ending positions and time.

Another way of writing the mathematical formula for velocity is:

\begin{align*} v\left(t\right) &= \ x'\left(t\right) \\ x\left(t\right) &= \int{v(t)\mathrm{d}t} \end{align*}

In words, this means the first derivative of an object’s position function gives the velocity function, and the integral of the velocity function gives the position function. These operations are both taken with respect to time. You’ll use these relationships to determine one function given the other.

Acceleration

We define the acceleration of an object undergoing motion in one dimension as follows.

We measure acceleration in units of length per squared unit of time, most commonly meters per second squared, or \(\frac{\mathrm{m}}{\mathrm{s}^2}\). Put simply, acceleration is changing velocity. This velocity change can be either speeding up or slowing down, or a directional change as well. The instantaneous acceleration, or acceleration at a specific moment in time, is:

$$ a_{x\,(\mathrm{inst})} = \frac{\mathrm{d}v}{\mathrm{d}t}. $$

To find the average acceleration over a period of time, we use the formula:

$$ a_{x\,(\mathrm{avg})} = \frac{v_f-v_0}{t_f-t_0} = \frac{\Delta v}{\Delta t}, $$

where \(\Delta v\) is the change in velocity. Finally, we can again write this relationship between position, velocity, and acceleration functions differently using calculus:

\begin{align*} a(t) &= v'(t) = x''(t) \\ v(t) &= \int{a(t)\mathrm{d}t} \\ x(t) &= \int{v(t)\mathrm{d}t} = \iint{a(t)\mathrm{d}t}. \end{align*}

In words, this says that the velocity function is the integral of the acceleration function, and the position function is the double integral of the acceleration function.

Describing Motion and Kinematics in One Dimension

Aside from the variables we just introduced, one of the most important tools for describing one-dimensional motion and kinematics are graphs. A few of the plots you’ll need to understand both how to interpret and create are:

• Position-time graphs, showing the distance traveled over time from the starting point.

• Velocity-time graphs, showing the changes in velocity over time.

• Acceleration-time graphs, showing the changes in acceleration over time.

Let’s take a brief look at each of these three types of graphs.

Next, let’s look at a velocity-time graph.

Finally, let’s examine an acceleration-time graph.

Graphs of motion in one dimension, along with some calculus, are powerful aids for understanding all sorts of motion. Now that we’ve walked through the tools we need, let’s look at a common problem type: vertical projectile motion in one dimension.

Vertical Projectile Motion in One Dimension

One of the first types of problems you’ll encounter is projectile motion.

An object thrown directly up in the air will accelerate downwards due to gravity, MikeRun via Wikimedia Commons CC BY-SA 4.0

In a vertical projectile motion problem, we throw an object like a ball up in the air, starting at an initial vertical position of \(h_0\). While tossing, we only give the object an initial vertical velocity \(v_{x,0}\); the horizontal component is zero. After the initial throw, the ball reaches a maximum height before accelerating back downwards due to gravity. This is just a brief introduction — we’ll go into the specifics of vertical projectile motion in one dimension later on.

Motion in One Dimension Examples

To finish our introduction to motion in one dimension, let’s walk through an example

Let’s walk through one more example, this time using the calculus definition of velocity.

Now that you’ve seen an introduction to motion in one dimension, your next step is to learn one-dimensional kinematics, or the study of motion without reference to the forces involved. We’ll be going into more kinematics formulas for motion in one dimension in the Kinematics Equations article, so read on!