Unit 1.4 | Applying Kinematic Equations in 1 Dimension Easily
- 32 minutes
- 4 formulas
- 24 questions
- 2 videos
Every where you look there is motion. People biking, planes moving, ants crawling.
In this lesson we will learn how to analyze this motion mathematically. It’s easier and more fun than you might initially think!
Unit 1 Breakdown
In this lesson you will learn:
- The “Big 5” Kinematic equations
- How to use the equations
- A simple framework to solve 1-D kinematic questions
- Applying the framework to solve interesting real-world problems
Introduction
After touching down, how long does it take a plane to stop?
You drop a penny from the two story building. At what speed does it hit the ground?
These are kinematic problems, also known as motion problems. Your job is to find one of the missing kinematic variables: displacement, initial velocity, final velocity, acceleration, or time.
If you like numbers, this is the lesson for you.
And if you hate math, don’t worry… it’s incredibly simple!
Understanding kinematic equations
The “big” 5 kinematic equations, listed below, will be used to solve all kinematic problems.
- [katex] v_f = v_0 +at [/katex]
- [katex] v_f^2 = v_0^2 + 2a\Delta x [/katex]
- [katex]\Delta x = v_0t + \frac{1}{2}at^2 [/katex]
- [katex]\Delta x = v_ft – \frac{1}{2}at^2 [/katex]
- [katex] \Delta x = \frac{1}{2} t (v_f + v_0) [/katex]
These equations involve the kinematic variables, which was covered in Lesson 1.2 | The kinematic variables.
Variables and their symbols
Let’s quickly recap what each variable means.
- Displacement, its variable is ‘[katex]\Delta x [/katex]’ (pronounced “delta x”), measured in meters.
- You might see this written as ‘[katex]s[/katex]’ in some textbooks.
- Velocity, its variable is ‘[katex]v[/katex]’, measured in meters/second.
Notice that velocity is split into two variables:
- initial velocity ‘[katex]v_0[/katex]’ (pronounced “v naught”)
- NOTE: Sometimes written as ‘[katex]v_i[/katex]’ or ‘[katex]u[/katex]’ in some textbooks
- final velocity ‘[katex]v_f[/katex]’ (pronounced “v final”)
- Acceleration, its variable is ‘[katex]a[/katex]’ and has units of meters/second2.
- Time is the last variable. Its symbol is ‘[katex]t[/katex]’, and it’s measured in seconds.
To recap, we have 5 kinematic variables in total listed in the chart below.
| Variable name | Variable Symbol | Units |
|---|---|---|
| Displacement | [katex]\Delta x [/katex] | [katex]m[/katex] |
| Initial velocity | [katex]v_0 [/katex] | [katex]\frac{m}{s}[/katex] |
| Final velocity | [katex]v_f[/katex] | [katex]\frac{m}{s}[/katex] |
| Acceleration | [katex]a[/katex] | [katex]\frac{m}{s^2}[/katex] |
| Time | [katex]t[/katex] | [katex]s[/katex] |
The “Big 5” equations
Although we have 5 kinematic variables, each equation only uses 4.
This means, that each equation doesn’t use exactly 1 variable.
For example, the chart below, shows equation 1 doesn’t use displacement ([katex]\Delta x[/katex]).
| Equation | Variable not in equation |
| [katex] v_f = v_0 + at [/katex] | [katex]\Delta x[/katex] |
| [katex] (v_f)^2 = (v_0)^2 + 2a\Delta x [/katex] | [katex]t[/katex] |
| [katex]\Delta x = v_0t + \frac{1}{2}at^2 [/katex] | [katex]v_f[/katex] |
| [katex]\Delta x = v_ft – \frac{1}{2}at^2 [/katex] | [katex]v_0[/katex] |
| [katex] \Delta x = \frac{1}{2} t (v_f + v_0) [/katex] | [katex]a[/katex] |
PRO TIP: Memorizing these five equations is a must in helping you solve problems rapidly!
1D Kinematic Problems
One-dimension (1D), refers to an object moving ONLY in one direction: either horizontally (left and right) OR vertically (up and down). We’ll cover two dimensional (2D) problems in the next lesson.
For now, take a look at this 1D problem: An airplane accelerates down a runway at [katex] 3 \, \frac{m}{s^2}[/katex] for [katex]20[/katex] seconds until it lifts off the ground. Determine the distance traveled before takeoff.
This might seem hard at first. So let’s use a framework to break down how to solve these questions.
Framework for kinematic problems
Think of a framework as a template for solving a specific type of problem. This will be the first of many frameworks you will learn in this course.
Here’s my simple framework for solving 1 dimension kinematic problems:
- Read the word problem and identify 4 kinematic variables. The problem will ALWAYS give you 3 variables, and ask you to solve for 1 variable
- Pick a kinematic equation. To make this easy, look for the variable the problem doesn’t even mention. Find the equation that also doesn’t have it, using the equation table from above.
- Plug and chug! Plug in all the given numbers and solve for the unknown variable
This might still sound a bit confusing. So let’s put it into practice.
Once you catch on, revisit this framework, and everything will make much more sense.
Applying the framework
Problem: An airplane accelerates down a runway at [katex] 3 \, \frac{m}{s^2}[/katex] for [katex]20[/katex] seconds until it finally lifts off the ground. Determine the distance traveled before takeoff.
- Identify the 4 variables in the problem: [katex]a = 3 \, m/s^2[/katex], [katex]t = 20 \, s[/katex], [katex]v_0 = 0 \, m/s[/katex], [katex]\Delta x = [/katex] what we need to find.
- Pick an equation: Notice that this question does not involve the ‘[katex]v_f[/katex]’ variable. Thus, we will pick the equation that does not have [katex]v_f[/katex] : [katex]\Delta x = v_0t + \frac{1}{2}at^2 [/katex]
- Plug the numbers into the equation and solve for [katex]\Delta x[/katex]:
- We find that [katex]\Delta x = 600 \, m[/katex].
PQ – Kinematics in the x and y directions
Use the framework to solve the following 1D kinematic problems for both the x and y directions.
Check answers by clicking the full version of each question.
Kinematics in the horizontal (x) direction
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Kinematics in the vertical (y) direction
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Challenge (Bonus) kinematics questions
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PS – 1D Kinematics
In this video we will solve a few problems using my simple 3 step kinematics’ framework. Hopefully, by the end of the video, you’ll see how simple 1D kinematic problems are!
Helpful problem solving tips
- After picking the correct kinematic equation, re-arrange the equation for the variable you are trying to solve for. Then you can plug in the numbers all at once.
- Label your variables using subscripts. For example if you are working in the horizontal direction, you can label, acceleration as [katex] a_x[/katex] and velocity as [katex] v_x[/katex]. This will make more sense once we start solving 2D problems
- Get good at finding “hidden variables.” For example, if a problem tells you a car is moving at constant velocity, they are technically also telling you that acceleration = zero.
- If a problem deals with an object in free fall, you can assume acceleration = [katex]9.81 \, \frac{m}{s^2}[/katex].
Lesson 1.5 Preview
In the next lesson we will apply our problem solving framework to objects moving in two dimensions. We often call these projectiles. Although this may seem harder it’s actually quite simple using the tricks we will show.
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This article was written by Jason