This post is a brief introduction to kinematics or the study of motion. We will discuss what kinematics is and introduce important ideas such as vectors and scalars, the difference between distance and displacement, the difference between speed and velocity, and a short description of acceleration.

What is Kinematics?

Kinematics might sound a bit scary, but it’s really just a fancy way of saying the motion of objects. The study of kinematics, therefore, is the study of motion. Kinematics units typically focus on common objects such as cars, books, and skydivers – things you can interact with. The ideas can be translated to the very small and the very large, but both scales require a bit of tweaking so we’ll stick to common objects for now.

The Difference Between Scalars and Vectors

Scalars and vectors are two different ways we can describe a given motion, but they have a key difference. A scalar only gives a magnitude (a number with some units) while a vector gives both a magnitude and a direction. While it may seem like vectors are better descriptors, making scalars useless, they both have their place in how we study motion. If you and a friend wanted to go pick up a pizza, they may ask how long the walk is and you would only need to tell them 5\text{ miles} – no direction necessary. If they ask you where the pizza place is, then you would need to say something like 5\text{ miles} North. In the first scenario, you gave your friend a scalar while in the second you gave them a vector. The difference between a vector and a scalar is an important one and you will see why they are both used in physics as you begin the study of motion.

Defining a Frame of Reference

A frame of reference is another important concept when talking about studying the motion of objects. When we talk about a frame of reference, we are talking about what an observer would see from a given position. Consider the van driving down the coast in the image. An observer standing on the left-hand side of the photo by the sign would see the van moving away. Meanwhile, an observer standing on the opposite side would see the van approaching. Being able to utilize a frame of reference allows us to better collect and share accurate data – especially when working with other scientists.

Position, Distance, and Displacement in Kinematics

The first thing we need to be able to do in kinematics is to describe where something is, where it’s been, and where it’s going. To complete these tasks, we have three terms – position, distance, and displacement. Position is likely a familiar word, though you might not use it often. Position refers to an object’s placement. If we return to our conversation about scalars and vectors, the pizza place has a position 5\text{ miles} North of your home. It may seem as though position could be given as just a scalar without a direction, but in Physics, we tend to choose to be overly precise and treat position as a vector.

Distance is also a familiar term, but it has a very specific definition in kinematics. Distance refers to the length of a path traveled by an object. This may feel obvious, but it’s actually more complicated than you may expect at first glance. If we consider the race track below, the distance refers to the length of the track multiplied by how many times a driver circles the track. Assuming the track is 100\text{ meters} long and the driver loops the track 10 times, their distance is 1{,}000\text{ meters}.

Displacement, on the other hand, refers to the distance between the starting and ending points for a given motion. Returning to our track below, if a driver circles the track 10 times, their displacement will be 0\text{ meters} because the finish line is the same as the starting line. There is no difference between the driver’s starting and final position.

Is Displacement a Vector a Scalar Quantity?

There is another key difference between distance and displacement – one is a vector and one is a scalar. Because distance accounts for the entire path taken, choosing a direction is not entirely possible. If you consider our race track above, you would have to include different directions for every turn and straightaway. Instead, we only take into account the magnitude of distance traveled, making it a scalar quantity.

Conversely, displacement looks only at starting and ending points. These will have a set distance between them in one direction from a given frame of reference. Because magnitude and direction are both important, displacement is considered a vector quantity.

Kinematic Formula for Distance

To find the distance and displacement of different movements, we have two equations to use. For distance, we add up the distances of each segment of a motion. Distance is typically measured in meters (symbolized by an \text{m}) and represented in equations by the symbol d.

Formula for Distance d_{T}=d_{1}+d_{2}+...+d_{n}

Here, d_{T} is the total distance traveled, and d_{1} and d_{2} are the first and second segments, respectively. In some problems, you may have only one or two distances to add together while in others you may have more. d_{n} represents the highest number segment you have to add. If you go back to our race track, there were 10 loops. That means we had 10 segments so, d_{n} would be d_{10}.

Kinematic Formula for Displacement

Displacement is still measured in meters, but it is represented by \Delta x. The \Delta symbol is called a delta and can be thought of as meaning “change in”. In the case of displacement, \Delta x means change in position. 

Formula for Displacement \Delta x=x_{f}-x_{i}

Here, \Delta x is the net displacement (total displacement), x_{f} is the final position, and x_{i} is the initial position. It is important to note that if your initial position has a higher value than your final position, your displacement can be negative. 

Depending on the level of physics course you are taking and what textbook your course is referencing, you may see only x or x_{T} to represent overall or total displacement.

Example 1: Distance and Displacement Going to School

Assume the image above shows the arrangement of your home, school, and favorite bakery. If a morning walk to school involves stopping at the bakery for a breakfast pastry, find your distance and displacement for the trip.

How to Find Distance

To find the distance you will need to account for the length of the entire trip – from the home, past the school, to the bakery, and then back to the school. In this case:

\text{distance}=60\text{ m}+30\text{ m}+30\text{ m}

\text{distance}=120\text{ m}

How to Find Displacement

To find the displacement you only need to look at how far apart the starting and ending points are. In this case, you started at home and ended at school. These two locations are 60\text{ m} apart so:

\text{displacement}=60\text{ m}

Example 2: Distance and Displacement of a Ball

If a ball is kicked toward a wall 3\text{ m} away and then bounces off and rolls to a spot 5\text{ m} behind where it was kicked, what were the ball’s distance and displacement? Assume to the right is positive.

How to Find Distance

To find the distance, start by picking out each segment of the ball’s path.

• d_{1}=\text{moving to the wall}=3\text{ m}
• d_{2}=\text{bouncing back to the original spot}=3\text{ m}
• d_{3}=\text{rolling behind the original spot}=5\text{ m}

Note that all of these values are positive even though some would be moving in what we had defined as the negative direction. Your distance should always come out as a positive number and the only way to guarantee that is by adding all positive numbers together. Now that we’ve found our numbers, we can plug them into our equation for total distance:

d_{T}=d_{1}+d_{2}+d_{3}

d_{T}=3\text{ m}+3\text{ m}+5\text{ m}

d_{T}=11\text{ m}

How to Find Displacement

To find displacement, start by picking out the final and initial positions of the ball.

• x_{f}=\text{where the ball stopped rolling}=-5\text{ m}
• x_{i}=\text{where the ball was kicked}=0\text{ m}

Note that the initial position is zero. This will generally be the case, though you may have times where your frame of reference will change that for some problems. It is also worth noting that, in this problem, we’re putting the frame of reference at where the ball was kicked so that it ends up behind the observer, making our final position negative. Now that we’ve found all of our values, it’s time to plug them in to find our displacement:

\Delta x=x_{f}-x_{i}=-5\text{ m}-0\text{ m}

\Delta x=-5\text{ m}

For those of you watching closely, you may have noticed that this value doesn’t appear to have a direction even though displacement is a vector. While this value doesn’t mention a cardinal direction, a coordinate plane, or even left and right, it does have a direction. We had defined motion to the right as positive. The negative symbol before the 5 tells us that the ball must be moving to the left.

Note that you may have noticed that the displacement in both of these problems was less than the distance in both of these problems. This will generally be true as displacement doesn’t take into account any of the twists and turns that distance does. 

Speed and Velocity in Kinematics

When studying the motion of objects, we may need to know more than just where they went – we may also need to know how quickly they got there. For this, we can refer to speed and velocity. These terms may both be rather familiar, but (like distance) they have very specific uses in physics. There are also some key differences between speed and velocity that you will see below.

Kinematic Formula for Speed

Speed specifically refers to how quickly a given distance is covered. In physics terms, we would say it is the rate of change of distance. As speed is based on distance, a scalar quantity, it is also a scalar quantity and only has a magnitude and no direction. As such, it will always be a positive value – like how speedometers on cars only have positive values. Speed is represented with an s in equations and is equal to the distance covered divided by the change in time.

Formula for Speed s=d/\Delta t

Here, \Delta t is the change in time, measured in seconds (represented by an s). The units for speed follow directly from the equation. The equation shows meters divided by seconds and the units for speed are, quite logically, meters per second (represented by \text{m/s}).

Kinematic Formula for Velocity

Velocity has a somewhat similar definition, but instead of being the rate of change of distance, it is the rate of change of displacement. Velocity measures how quickly an object changed its position, regardless of how long the path was. Because velocity is based upon displacement, a vector quantity, velocity is also a vector quantity with both a magnitude and a direction. Velocity is represented by a v in equations and is equal to the displacement divided by the change in time.

Formula for Velocity v=\Delta x/\Delta t

Here again, \Delta t is the change in time. Similar to speed, velocity units can be derived from the equation. Displacement is measured in meters while time is measured in seconds so meters divided by seconds give us meters per second (still represented by \text{m/s}).

Note: Depending on the level of physics course you are taking and what textbook your course is referencing, you may only see t to represent time rather than \Delta t.

Example 1: Speed and Velocity Going to School

Let’s revisit the trip to school with a stop at the bakery. We already know this trip had a distance of 120 \text{ m} and a displacement of 60\text{ m}. Let’s also assume the trip took 600\text{ seconds} (10\text{ minutes}). Find our speed and velocity.

How to Find Speed

For speed, we’ll start by pulling out the values we need – distance and time:

• d=120\text{ m}
• \Delta t=600\text{ s}

Now we can plug these values into our speed equation and solve.

s=d/\Delta t

s=120\text{ m}/600\text{ s}

s=0.2\text{ m/s}

How to Find Velocity

Velocity starts the same way, but instead of pulling out distance and time, we’ll pull out displacement and time.

• \Delta x=60\text{ m}
• \Delta t=600\text{ s}

Now we can plug these values into our velocity equation and solve just as we did before.

v=\Delta x/\Delta t

v=60\text{ m}/600\text{ s}

v=0.1\text{ m/s}

Note, the velocity in this problem is less than the speed. This will generally be true as displacement is generally less than distance.

Example 2: Speed and Velocity of a Ball

Now let’s look at a ball rolling down a hill. The hill is 50\text{ m} and it takes 25\text{ s} for the ball to finish rolling down the hill. Assume that up is positive.

How to Find Speed

For speed, we’ll start by pulling out the values we need – distance and time.

• d=50\text{ m}
• \Delta t=25\text{ s}

Now we can plug these values into our speed equation and solve:

s=d/\Delta t

s=50\text{ m}/25\text{ s}

s=2\text{ m/s}

How to Find Velocity

Velocity starts the same way, but instead of pulling out distance and time, we’ll pull out displacement and time.

• \Delta x=-50\text{ m}
• \Delta t=25\text{ s}

Now we can plug these values into our velocity equation and solve just as we did before:

v=\Delta x/\Delta t

v=-50\text{ m}/25\text{ s}

v=-2\text{ m/s}

Note that speed and velocity have the same magnitude in this problem. The velocity is negative to show the direction of motion being down the hill, but the value of 2\text{ m/s} applies to both the speed and the velocity. This will be true whenever your distance and displacement are the same value.

Acceleration in Kinematics

Sometimes we need to go a step beyond finding how quickly something moves. For example, we may need to know how long it takes it to reach a certain speed – either speeding up or slowing down. The rate of change for both speed and velocity is called acceleration. Acceleration can be either a scalar or a vector depending on whether you are using it for speed or velocity. If you are using acceleration as the rate of change of speed it will be a scalar because speed is a scalar. Similarly, if you are using acceleration as the rate of change of velocity it will be a vector because velocity is a vector. Acceleration is important to kinematics as well as a wide variety of other topics in physics.

Tips for Solving Kinematics Problems

While a problem with a visual makes it easy to find the values you need, you will also need to be able to find these values from physics word problems. When looking through a physics word problem there are a few steps you will want to take each time. 

1. Identify what the question is asking (Does it want a distance? A displacement? Both?). 
2. Pull out any useful information for solving the problem (Where are things placed? How far apart are they? What direction have they gone?).
3. Find any equations you can use to solve the problem. 
4. Plug in your values and solve.

Understanding and Practicing Kinematics

Kinematics introduces many of the fundamental principles of physics. As you continue through physics, you will likely find it difficult to escape the ideas of displacement and velocity and even units. If you can get comfortable with distance, displacement, speed, velocity, and working with these equations, you’ll have a strong base for moving forward with the rest of your kinematics unit.