Kinematics Physics

The Basic Concepts of Kinematics

Before we introduce the key kinematics equations, let’s briefly go through the background information and various parameters you need to know first.

Scalars and Vectors

In kinematics, we can divide physical quantities into two categories: scalars and vectors.

In other words, a scalar is simply a numerical measurement with a size. This can be a plain old positive number or a number with a unit that doesn’t include a direction. Some common examples of scalars that you regularly interact with are:

• The mass (but not weight!) of a ball, textbook, yourself, or some other object.

• The volume of coffee, tea, or water contained in your favorite mug.

• The amount of time passed between two classes at school, or how long you slept last night.

So, a scalar value seems pretty straightforward — how about a vector?

When we say that a vector has direction, we mean that the direction of the quantity matters. That means the coordinate system we use is important, because the direction of a vector, including most variables of kinematic motion, will change signs depending on whether the direction of motion is positive or negative. Now, let’s look at a few simple examples of vector quantities in daily life.

• The amount of force you use to push open a door.

• The downward acceleration of an apple falling from a tree branch due to gravity.

• How fast you ride a bike east starting from your home.

You’ll encounter several conventions for denoting vector quantities throughout your physics studies. A vector can be written as a variable with a right arrow above, such as the force vector \(\overrightarrow{F}\) or a bolded symbol, such as \(\mathbf{F}\). Make sure you’re comfortable working with multiple types of symbols, including no denotation for vector quantities!

Variables in Kinematics

Mathematically solving kinematics problems in physics will involve understanding, calculating, and measuring several physical quantities. Let’s go through the definition of each variable next.

Position, Displacement, and Distance

Before we know how fast an object is moving, we have to know where something is first. We use the position variable to describe where an object resides in physical space.

For simple linear motion, we use a one-dimensional axis, such as the \(x\), \(y\), or \(z\)-axis. For motion along the horizontal axis, we denote a position measurement using the symbol \(x\), the initial position using \(x_0\) or \(x_i\), and the final position using \(x_1\) or \(x_f\). We measure position in units of length, with the most common unit choice being in meters, represented by the symbol \(\mathrm{m}\).

If we instead want to compare how much an object’s final position differs from its initial position in space, we can measure the displacement after an object has undergone some type of linear motion.

We measure the displacement \(\Delta x\), sometimes denoted as \(s\), using the same units as position. Sometimes, we only want to know how much ground an object has covered altogether instead, such as the total number of miles a car has driven during a road trip. This is where the distance variable comes in handy.

In other words, we sum up the absolute value of the length of each segment along a path to find the total distance \(d\) covered. Both displacement and distance are also measured in units of length.

Displacement measurements describe how far an object has moved from its starting position, while distance measurements sum up the total length of the path taken, Stannered via Wikimedia Commons CC BY-SA 3.0

The most important distinction to remember between these quantities is that position and displacement are vectors, while distance is a scalar.

Since displacement calculations take direction into account, these measurements can be positive, negative, or zero. However, distance can only be positive if any motion has occurred.

Time

An important and deceptively simple variable that we rely on for both day-to-day structure and many physics problems is time, particularly elapsed time.

We measure a time interval \(\Delta t\) as the difference between the final timestamp and initial timestamp, or:

\begin{align*} \Delta t=t_f-t_i \end{align*}

We record time typically in units of seconds, denoted by the symbol \(\mathrm{s}\) in physics problems. Time may seem very straightforward on the surface, but as you journey deeper into your physics studies, you’ll find that defining this parameter is a bit more difficult than before! Don’t worry — for now, all you need to know is how to identify and calculate how much time has passed in a problem according to a standard clock or stopwatch.

Velocity and Speed

We often talk about how “fast” something is moving, like how fast a car is driving or how quickly you’re walking. In kinematics, the concept of how fast an object is moving refers to how its position is changing through time, along with the direction it’s headed.

In other words, the velocity variable \(v\) describes how much an object changes its position for each unit of time that passes. We measure velocity in units of length per time, with the most common unit being in meters per second, denoted by the symbol \(\mathrm{\frac{m}{s}}\). For example, this means that an object with a velocity of \(10\,\mathrm{\frac{m}{s}}\) moves \(\mathrm{10\, m}\) every second that passes.

Speed is a similar variable, but instead calculated using the total distance covered during some period of elapsed time.

We measure the speed \(s\) using the same units as velocity. In everyday conversation, we often use the terms velocity and speed interchangeably, whereas in physics the distinction matters. Just like displacement, velocity is a vector quantity with direction and magnitude, while speed is a scalar quantity with only size. A careless mistake between the two can result in the wrong calculation, so be sure to pay attention and recognize the difference between the two!

Acceleration

When driving a car, before we reach a constant speed to cruise at, we have to increase our velocity from zero. Changes in the velocity result in a nonzero value of acceleration.

In other words, acceleration describes how quickly the velocity changes, including its direction, with time. For example, a constant, positive acceleration of \(indicates a steadily increasing velocity for each unit of time that passes.

We use units of length per squared time for acceleration, with the most common unit being in meters per second squared, denoted by the symbol \(\mathrm{\frac{m}{s^2}}\). Like displacement and velocity, acceleration measurements can be positive, zero, or negative since acceleration is a vector quantity.

Forces

You likely already have enough physical intuition to guess that motion can’t simply occur from nothing — you have to push your furniture to change its position when redecorating or apply a brake to stop a car. A core component of motion is the interaction between objects: forces.

Forces are vector quantities, which means the direction of the interaction is important. Force measurement can be positive, negative, or zero. A force is usually measured in units of Newtons, denoted by the symbol \(\mathrm{N}\), which is defined as:

\begin{align*} \mathrm{1\, N=1\,\frac{kg\cdot m}{s^2}}\end{align*}

According to our definition of kinematics, we don’t need to account for any pushing or pulling interactions that might’ve kick-started motion. For now, all we need to pay attention to is the motion as it’s happening: how fast a car is traveling, how far a ball has rolled, how much an apple is accelerating downward. However, it’s beneficial to keep forces such as gravity in the back of your mind as you analyze kinematics problems. Kinematics is just a stepping stone to building our understanding of the world before we dive into more difficult concepts and systems!

Kinematic Equations in Physics

The kinematics equations, also known as equations of motion, are a set of four key formulas we can use to find the position, velocity, acceleration, or time elapsed for the motion of an object. Let’s walk through each of the four kinematic equations and how to use them.

The first kinematic equation allows us to solve for the final velocity given an initial velocity, acceleration, and time period:

\begin{align*} v=v_0+a \Delta t \end{align*}

where \(v_0\) is the initial velocity, \(a\) is the acceleration, and \(\Delta t\) is the time elapsed. The next kinematic equation lets us find the position of an object given its initial position, initial and final velocities, and elapsed time:

\begin{align*} x=x_0+(\frac{v+v_0}{2}) \Delta t,\, \mathrm{or} \\ \Delta x=(\frac{v+v_0}{2}) \Delta t \end{align*}

where \(x_0\) is the initial position in the \(x\)-direction. We can substitute \(x\) for \(y\) or \(z\) for motion in any other direction. Notice how we’ve written this equation in two different ways — since the displacement \(\Delta x\) is equal to \(x-x_0\), we can move our initial position variable to the left side of the equation and rewrite the left side as the displacement variable. This handy trick also applies to our third kinematic equation, the equation for the position given the initial position, initial velocity, acceleration, and elapsed time:

\begin{align*} x=x_0+v_0t+\frac{1}{2}a\Delta t^2,\, \mathrm{or} \\ \Delta x=v_0t+\frac{1}{2}a\Delta t^2 \end{align*}

Again, we can always substitute the position variables with whichever variable we need in a given problem. Our final kinematic equation allows us to find the velocity of an object with only the initial velocity, acceleration, and displacement:

\begin{align*} v^2=v_0^2+2a\Delta x \end{align*}

All four of the kinematic equations assume that the acceleration value is constant, or unchanging, during the time period we observed the motion. This value could be the acceleration due to gravity on the surface of Earth, another planet or body, or any other value for acceleration in another direction.

Choosing which kinematic equation to use might seem confusing at first. The best method to determine which formula you need is by listing the information you’ve been given in a problem by variable. Sometimes, the value of a variable may be implied in the context, such as zero initial velocity when dropping an object. If you think you haven’t been given enough details to solve a problem, read it again, and draw a diagram too!

Types of Kinematics

Though kinematics in physics broadly includes motion without regard to causal forces, there are several types of recurring kinematics problems you’ll encounter as you begin your studies of mechanics. Let’s briefly introduce a few of these types of kinematic motion: free fall, projectile motion, and rotational kinematics.

Free Fall

Free fall is a type of one-dimensional vertical motion where objects accelerate only under the influence of gravity. On Earth, the acceleration due to gravity is a constant value we represent with the symbol \(\mathrm{g}\):

\begin{align*} \mathrm{g=9.81\, \frac{m}{s^2}} \end{align*}

Free fall motion occurs in only the vertical direction, starting at a height h naught above the ground, MikeRun via Wikimedia Commons CC BY-SA 4.0

In the case of free fall, we don’t consider the effects of air resistance, friction, or any initially applied forces that don’t fit in with the definition of free-falling motion. An object undergoing free fall motion will descend a distance of \(\Delta y\), sometimes called \(\mathrm{h_0}\), from its initial position to the ground. To get a better understanding of how free fall motion works, let’s walk through a brief example.

Though most free fall problems take place on Earth, it’s important to note that acceleration due to gravity on different planets or smaller bodies in space will have different numeric values. For example, acceleration due to gravity is considerably smaller on the moon and significantly greater on Jupiter than what we’re used to on Earth. So, it isn’t a true constant — it’s only “constant” enough for simplifying physics problems on our home planet!

Projectile Motion

Projectile motion is the two-dimensional, usually parabolic motion of an object that has been launched into the air. For parabolic motion, an object’s position, velocity, and acceleration can be split into horizontal and vertical components, using \(x\) and \(y\) subscripts respectively. After splitting a variable of motion into individual components, we can analyze how fast the object moves or accelerates in each direction, as well as predict the position of the object at different points in time.

An object with projectile motion launched at an angle will have velocity and acceleration in both the x and y directions, StudySmarter Originals

All objects experiencing projectile motion exhibit symmetric motion and have a max range and height — as the classic saying goes, “what goes up must come down”!

Rotational Motion

Rotational motion, also known as rotational kinematics, is an extension of the study of linear kinematics to the motion of orbiting or spinning objects.

Examples of rotational motion exist all around us: take the planetary orbits revolving around the Sun, the inner movement of cogs in a watch, and the rotation of a bicycle wheel. The equations of motion for rotational kinematics are analogous to the equations of motion for linear motion. Let’s look at the variables we use to describe rotational motion.

Kinematics and classical mechanics as a whole are extensive branches of physics that may feel daunting at first. But don’t worry — we’ll be going into much more detail for all the new variables and equations in the next few articles!