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Unit 1.4 | Applying Kinematic Equations in 1 Dimension Easily

Jason Kuma
Jason Kuma

Writer | Coach | USC - Physics B.S & Business B.A. | Fremont, CA

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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

You are on Lesson 4 of 5

  1. Unit 1.1 | Understanding vectors and the Standard Units used in Physics
  2. Unit 1.2 | The Kinematic (motion) variables: Displacement, Velocity, and Acceleration
  3. Unit 1.3 | Graphing motion
  4. Unit 1.4 | Using Kinematic Equations in 1 Dimension [Current Lesson]
  5. Unit 1.5 | Projectile Motion: Using Kinematic Equations in 2 Dimensions

In this lesson you will learn: 

  • The “Big 4” 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


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” 4 kinematic equations, listed below, will be used to solve all kinematic problems.

  1. v_f = v_0 +at
  2. v_f^2 = v_0^2 + 2a\Delta x
  3. \Delta x = v_0^2 + \frac{1}{2}at^2
  4. \Delta x = \frac{1}{2} t (v_f + v_0)

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.

  1. displacement, it’s variable is ‘∆x’ (pronounced “delta x”) NOTE: You might see this written as ‘s’ in some textbooks. It has units of meters
  2. velocity, it variable is ‘v’. It has units of meters/second.

Notice that velocity is split into two variables:

  • initial velocity ‘v0 (pronounced “v knot”) NOTE: You might also see this written also as ‘vi‘ or even ‘u’ in some textbooks
  • final velocity ‘vf (pronounced “v final”)
  1. Acceleration, its variable is ‘a’. It has units of meters/second2.
  2. Time is the final variable. Its symbol is ‘t’, and its measured in seconds.

So to recap, we have 5 kinematic variables in total listed in the chart below.

Variable nameVariable SymbolUnits
Initial velocityvom/s
Final velocity vfm/s
Time ts

The “Big 4” 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, equation 1, on the chart above, is missing (does not use) the acceleration variable.

PRO TIP: Memorizing these 4 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).

 In the next lesson, we’ll cover two dimensional (2D) problems. 

Let’s take this problem for example: An airplane accelerates down a runway at 3.20 m/s2 for 32.8 s until is finally lifts off the ground. Determine the distance traveled before takeoff. 

This might seem hard at first. So lets break it down using the framework below.

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:

  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 3 m/s2 for 20 seconds until is finally lifts off the ground. Determine the distance traveled before takeoff.

  1. Identify the 4 variables in the problem: a = 3 m/s2, t = 20 seconds, v0 = 0 m/s, ∆x = what they want us to find.
  2. Pick an equation: Notice that this question does not involve the ‘vf‘ variable. Thus, we will pick the equation that does not have vf : ∆x = v0t +1/2at2
  3. Plug the numbers into the equation and solve for ∆x: We find that ∆x = 600 meters

PQ – Kinematics in the x and y directions

Using the framework see if you can solve the following problems.

The problems are split into two types. Problems in the horizontal direction. And problems in the vertical direction. Both are 1-D kinematics, as it only involved motion in ONE direction.

Answers are given in (bold text).

Kinematics in the horizontal (x) direction

  1. An airplane accelerates down a runway at 10 m/s2. It reaches a final velocity of 200 m/s until is finally lifts off the ground. Determine the distance traveled before takeoff. (2000 m)
  2. 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. (8 m/s2)
  3. A car decelerates from 25 m/s to 5 m/s at 10 m/s2. How far does the car travel during this deceleration? (30 m)
  4. A car traveling at 20 m/s decelerates at a constant rate to a complete stop after traveling 40 m. (a) What is the average speed of the car during this process? (10 m/s). (b) How long does it take for the car to stop? (4 seconds)

Kinematics in the vertical (y) direction

  1. A ball is dropped from a window 10 m above the sidewalk. Determine the time it takes for the ball to fall to the sidewalk. (1.4 seconds)
  1. A tennis ball is thrown straight up with an initial speed of 22.5 m/s. It is caught at the same distance above ground.
  1. How high does the ball rise? (25.3 m)
  2. How long does it take for the ball to reach its highest point? (2.25 seconds)
  3. How long does the ball remain in the air? (4.5 seconds)
  4. How fast was it going just before it is caught? (22.5 m/s)
  5. What is the velocity and acceleration of the ball at the highest point? (0 m/s, -10m/s2)
  1. A rock is thrown vertically upward with a velocity of 20 m/s from the edge of a bridge 42 m above a river. (a) What is the rock’s speed just before it falls into the river? (35.2 m/s). (b) How much time does it take from the time the rock is launched to the time when the rock strikes the river water? (5.5 seconds)

Challenge (Bonus) kinematics questions

  1. A car is traveling at 18 m/s when the driver sees a disabled car in the middle of the road. He takes 0.8 s to react (assume that the car travels at constant speed during this reaction time), then steps on the brakes and slows at 9.0 m/s2. How far does the car go before it stops? (32.4 m)
  2. 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. (a) What is the difference in the time the balls spend in the air? (3 s). (b) What is the velocity of each ball as it strikes the ground? (-24.5 m/s). (c) How far apart are the balls 0.800 s after they are thrown? (23.5 m)

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 ax and velocity as vx. 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 = 9.81 m/s2.

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. 

Jason Kuma
Jason Kuma

Writer | Coach | USC - Physics B.S & Business B.A. | Fremont, CA

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Units in AP Physics 1

Unit 1 – Linear Kinematics

Unit 2 – Linear Forces

Unit 3 – Circular Motion

Unit 4 – Energy 

Unit 5 – Momentum 

Unit 6 – Torque 

Unit 7 – Oscillations 

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