--- title: "Unit 1.4 | Applying Kinematic Equations in 1 Dimension Easily" description: "Learn how to use the kinematic equations a simple way to solve kinematic problems in 1 dimension. Beginners, AP Physics 1 + all curriculums." featured_image: "https://nerd-notes.com/wp-content/uploads/2023/06/max-bender-YOk0D5sz3e4-unsplash-1-1.jpg" url: "https://nerd-notes.com/unit-1-3-mastering-motion-applying-kinematic-equations-in-1-dimension/" date_modified: "2024-09-04T03:48:19+00:00" --- # Unit 1.4 | Applying Kinematic Equations in 1 Dimension Easily ## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-1-3-mastering-motion-applying-kinematic-equations-in-1-dimension/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-1-3-mastering-motion-applying-kinematic-equations-in-1-dimension/ ### Unit 1 Breakdown **You are on Lesson 4 of 5** 1. [Unit 1.1 | Understanding vectors and the Standard Units used in Physics](https://nerd-notes.com/unit-1-1-understanding-vectors-and-the-standard-units-used-in-physics/) 2. [Unit 1.2 | The Kinematic (motion) variables: Displacement, Velocity, and Acceleration](https://nerd-notes.com/unit-1-mastering-motion-a-beginners-guide-to-kinematics-in-ap-physics-1/) 3. [Unit 1.3 | Graphing motion](https://nerd-notes.com/unit-1-2-mastering-motion-graphing-kinematic-variables/) 4. [Unit 1.4 | Using Kinematic Equations in 1 Dimension [Current Lesson]](https://nerd-notes.com/unit-1-3-mastering-motion-applying-kinematic-equations-in-1-dimension/) 5. [Unit 1.5 | Projectile Motion: Using Kinematic Equations in 2 Dimensions](https://nerd-notes.com/unit-1-4-projectile-motion-motion-in-2-dimensions/) #### 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 the one of 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. 1. [katex] v_f = v_0 +at [/katex] 2. [katex] v_f^2 = v_0^2 + 2a\Delta x [/katex] 3. [katex]\Delta x = v_0t + \frac{1}{2}at^2 [/katex] 4. [katex]\Delta x = v_ft – \frac{1}{2}at^2 [/katex] 5. [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.](https://nerd-notes.com/unit-1-mastering-motion-a-beginners-guide-to-kinematics-in-ap-physics-1/) #### Variables and their symbols Let’s quickly recap what each variable means. 1. **Displacement**, it’s variable is **‘**[katex]\Delta x [/katex]’ (*pronounced “delta x”)*, measured in meters. - You might see this written as ‘[katex]s[/katex]’ in some textbooks. 2. **Velocity**, it 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 knot”)* - **NOTE:** Sometimes written as ‘[katex]v_i[/katex]’ or ‘[katex]u[/katex]’ in some textbooks - **final velocity** ‘[katex]v_f[/katex]’ *(pronounced “v final”)* 1. **Acceleration**, its variable is ‘[katex]a[/katex]’ and has units of meters/second2. 2. **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] | *Five kinematic variables and their symbols.* #### The “Big 5” equations Although we have 5 kinematic variables, each equation only uses 4.  This means, that each equation is 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] | *Five kinematic equations and the missing variable in each.* **PRO TIP:** Memorizing these five equations are a must in helping you solve problems rapidly! ### 1D Kinematic Problems One-dimension (1D), refers to an object moving in 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](https://nerd-notes.com/kinematic-framework-solving-hard-projectile-problems-the-easy-way/) 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: 1. **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 2. **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. 3. **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 in 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 is finally lifts off the ground. Determine the distance traveled before takeoff. 1. 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. 2. 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] 3. 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 Using the framework to solve 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 [Open in UBQ](https://nerd-notes.com/ubq/40633/) Question 1 Beginner Mathematical An airplane accelerates down a runway at \( 10 \, \text{m/s}^2 \). It reaches a final velocity of \( 200 \, \text{m/s} \) until it finally lifts off the ground. Determine the distance traveled before takeoff. Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/40634/) Question 2 Beginner Mathematical A car starts from rest and accelerates uniformly over a time of 5 seconds for a distance of 100 m. Determine the acceleration of the car. Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/40635/) Question 3 Beginner Mathematical A car decelerates from \( 25 \, \text{m/s} \) to \( 5 \, \text{m/s} \) at \( 10 \, \text{m/s}^2 \). How far does the car travel during this deceleration? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/40636/) Question 4 Beginner Mathematical A car traveling at \( 20 \, \text{m/s} \) decelerates at a constant rate to a complete stop after traveling \( 40 \, \text{m} \). Part (a)3 pts What is the average speed of the car during this process? Part (b)3 pts How long does it take for the car to stop? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) #### Kinematics in the vertical (y) direction [Open in UBQ](https://nerd-notes.com/ubq/40648/) Question 5 Beginner Mathematical A ball is dropped from a window \(10 \, \) above the sidewalk. Determine the time it takes for the ball to fall to the sidewalk. Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/40649/) Question 6 Beginner Mathematical A tennis ball is thrown straight up with an initial speed of \( 22.5 \, \text{m/s} \). It is caught at the same distance above ground. Part (a)3 pts How high does the ball rise? Part (b)3 pts How long does it take for the ball to reach its highest point? Part (c)3 pts How long does the ball remain in the air? Part (d)3 pts How fast (speed) was it going just before it is caught? Part (e)3 pts What is the velocity and acceleration of the ball at the highest point? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/40650/) Question 7 Beginner Mathematical A rock is thrown vertically upward with a velocity of \( 20 \, \text{m/s} \) from the edge of a bridge \( 42 \, \text{m} \) above a river. Part (a)3 pts What is the rock’s speed just before it falls into the river? Part (b)3 pts How much time does it take from the time the rock is launched to the time when the rock strikes the river water? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) #### Challenge (Bonus) kinematics questions [Open in UBQ](https://nerd-notes.com/ubq/19130/) Question 8 Advanced Mathematical A car is traveling 20 m/s when the driver sees a child standing on the road. She takes 0.8 s to react then steps on the brakes and slows at 7.0 m/s2. How far does the car go before it stops? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/19109/) Question 9 Advanced Mathematical Two students are on a balcony 19.6 m above the street. One student throws a ball vertically downward at 14.7 m/s. At the same instant, the other student throws a ball vertically upward at the same speed. The second ball just misses the balcony on the way down. Part (a)2 pts What is the difference in the time the balls spend in the air? Part (b)2 pts What is the velocity of each ball as it strikes the ground? Part (c)3 pts How far apart are the balls 0.800 s after they are thrown? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) ### PS – 1D Kinematics In this [video](https://www.youtube.com/watch?v=E51VATYMheU) 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!  [Embedded content](https://www.youtube.com/embed/E51VATYMheU?feature=oembed) ### 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](https://nerd-notes.com/unit-1-4-projectile-motion-motion-in-2-dimensions/) 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.  ## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-1-3-mastering-motion-applying-kinematic-equations-in-1-dimension/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-1-3-mastering-motion-applying-kinematic-equations-in-1-dimension/