Kinematics is the study of motion. One-dimensional kinematics describes an object moving along a line (or in 1-D).

When approaching a kinematics problem, it is helpful to draw yourself a diagram of the problem and label all the quantities which are known and the one you need to find. Once you know what you have and what you are looking for, you can try to find equations which connect these quantities.

Check out an example of a 1-D kinematics problem: The Case of the Falling Grapefruit

Sometimes thinking through where the kinematics equations come from can help us to learn these equations and be able to recall them more quickly when we need to use them on a quiz or exam. This post includes the derivations of two 1-D kinematics equations. Note that these derivations use both integrals and derivatives. I don’t recommend viewing the derivations if you haven’t had calculus yet.

Topics in One-Dimensional Kinematics:

Velocity vs. Speed
Acceleration
Constant Acceleration
Falling Objects

Simulations

Interactive Velocity Exercise from The Physics Classroom

Kinematics problems involve an object (or objects) in motion and these generally come in the form of word problems. Let’s go through an example together. The word problem we will be looking at is written below.

Word problem: Longtime friends, Maureen and Roger, are having a picnic on the rooftop of their apartment building. A grapefruit gets accidentally dropped from the rooftop and falls straight down to the ground. The falling grapefruit falls past their neighbor Paloma’s window. The grapefruit takes 0.125 s to fall from the top to the bottom of Paloma’s window, a distance of 1.20 m. From the bottom of the window it takes 1.0 s for the grapefruit to hit the ground. How tall is the apartment building?

Reading and setting up the problem: Watch the video below to see a walkthrough of reading the problem and determining the following:

• The goal (or what variable we are being asked to solve for)
• The variables/information given
• How to diagram the information
• Determining potentially useful equations

Now that you’ve setup the problem, see if you can come with a problem solving strategy to determine Δx, the height of the apartment building. Once you’ve thought about your possible strategy, check out the two videos below for a couple of strategies that we came up with.

Note that everyone has different problem solving experiences, so our thought processes and thus problem solving strategies might be different. For this particular problem (and for many problems) several valid problem solving strategies exist.

Strategy 1:

Strategy 2:

Sometimes it can help to anchor equations into the brain to know where the equation comes from. Added bonus, if you can’t remember whether or not there’s a negative sign or maybe a $\frac{1}{2}$ somewhere in the equation, knowing where the equation comes from provides you a way to think through it and figure this information out.

Below are a couple of quick derivations for two kinematics equations:

Video showing the derivation of $v = v_0 + at$ from the definition of acceleration:

Video showing the derivation of $\Delta x = v_0 t + \frac{1}{2} a t^2$: