The Kinematics Equations

Alright. We’ve defined our main concepts (displacement, velocity, and acceleration) and we’ve reviewed our algebra skills. It’s time to tackle the kinematics equations.

The whole point of the kinematics equations is that they mathematically describe motion. Note: they do not explain motion! We’ll worry about explaining motion later because explaining why things happen is always more difficult than merely observing that they do, in fact, happen.

The starting point for deriving (fancy physics/math speak for “developing”) the kinematics equations is an assumption that we’re going to make about our moving objects: we’re going to assume that they’re moving at constant (unchanging) acceleration*. We can guarantee that our acceleration is constant when the average acceleration equals the instantaneous acceleration at all points in time. Since it’s always the same, we can dispense with the subscripts and simply say that the object we’re considering moves with constant acceleration .

We’ve talked about acceleration and how it represents a change in an object’s velocity over time. Let’s mathematically unpack that definition a bit:

This last line is our first kinematics equation! It relates the starting (initial) velocity , the ending (final) velocity , the acceleration (remember that this is a constant vector, unchanging in both magnitude and direction), and the amount of time an object is traveling (time interval), . This may not feel like a very big accomplishment yet, but it is paving the way for more complicated equations.

Note that we cannot unpack the velocity in the same way because if there is an acceleration, the velocity is changing, and our unpacking of acceleration above assumed a constant acceleration. Only if the acceleration was zero (and velocity therefore constant) could we similarly write .

So if we can’t do that, what can we do? Well, let’s unpack the definition of average velocity. We know from our physics definition that . But there’s also the mathematical definition of the average of two numbers**: add them together and divide by 2! When we’re talking about average velocity in a mathematical sense, we could be talking about the average of the initial and final velocities, or . So we have two different expressions each equal to the average velocity. This means that these expressions must be equal to each other! (See my dad’s post on the Laws of Algebra.) Let’s mush these expressions together and rearrange them a bit:

This is our second kinematics equation! It relates our displacement () to how fast we’re going and how long we’re going that fast – no acceleration needed! (Though underlying this equation is of course the assumption that the acceleration is constant.)

Two down, two to go! Alright, let’s take stock of the variables in play here. We’ve got: , and . We have two equations: the first based on initial and final velocity, time, and acceleration; the second based on displacement, initial and final velocity, and time. Right now, we don’t have an equation directly relating displacement to acceleration, and that sure would be nice because often, we know our acceleration and would like to learn how far it got us. We can actually accomplish this goal in two different ways. I’ll show you the first and leave the second to you as an assignment. To begin, we’re going to substitute our first kinematics equation into the final velocity term in the second equation. We’re plugging

into

and we find that

This is our third kinematics equation! The fourth and final kinematics equation relates displacement, initial and final velocity, and acceleration in one equation. This one you’re going to derive yourself – but the idea is similar to what I did above. Instead of substituting from one equation to another, you’re going to solve one of the original two equations for and substitute this into one of the remaining equations.

There is, however, one thing to note: for this last equation, we will be dropping the vector hats because the concept of the “dot product” of two vectors is beyond the scope of this blog at present, and the dot product is what is necessary to correctly write this last equation generally for three-dimensional vectors. This means that you will combine 2 of the following 3 one-dimensional equations to derive your 4th kinematics equation:

Note that it does not matter which two equations you choose to do this final derivation, and I encourage you to pair them in various ways to prove to yourself that you get the same final equation every time. Behind the picture of Cheyenne is the solution based on combining the first two equations.

And that’s it! We have four equations to help us figure out how far objects have traveled, how fast they started out, how fast they ended, their acceleration, and how much time they were in motion. I’m going to reorder them slightly here and number them for future reference:

1. 
2. 
3. 
4. 

In my next post, we will talk about graphing equations (1) and (2) to take a deeper look at what they tell can tell us.

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*We could mathematically deal with a changing acceleration, but this, as we will learn, would require a changing force, and to properly deal with these things, we’d need calculus, and right now, this is an algebra-based blog. (Though when we’re done with the algebra version of physics, we’ll dive in to the calculus version!)

**We are allowed to use this straightforward definition of the average because our acceleration is constant, therefore making our velocity “linear.” This just means that if we graph the object’s velocity vs. time, the graph looks like a straight line. We’ll talk more about graphing motion in my next post.