# Unit 1.4 | Applying Kinematic Equations in 1 Dimension Easily

###### Jason Kuma

Writer | Coach | Builder | Fremont, CA

###### Article Content
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

#### 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” 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 \Delta x ‘ (pronounced “delta x”), measured in meters.
• You might see this written as ‘s‘ in some textbooks.
2. Velocity, it variable is ‘v‘, measured in meters/second.

Notice that velocity is split into two variables:

• initial velocityv_0(pronounced “v knot”)
• NOTE: Sometimes written as ‘v_i‘ or ‘u‘ in some textbooks
• final velocityv_f(pronounced “v final”)
1. Acceleration, its variable is ‘a‘ and has units of meters/second2.
2. Time is the last variable. Its symbol is ‘t‘, and it’s measured in seconds.

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

#### 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 (\Delta x).

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 3 \, \frac{m}{s^2} for 20 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:

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 \, \frac{m}{s^2} 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/s^2, t = 20 \, s, v_0 = 0 \, m/s, \Delta x = what we need to find.
2. Pick an equation: Notice that this question does not involve the ‘v_f‘ variable. Thus, we will pick the equation that does not have v_f : \Delta x = v_0t + \frac{1}{2}at^2
3. Plug the numbers into the equation and solve for \Delta x:
• We find that \Delta x = 600 \, m.

### 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

Question 1
Difficulty - Beginner
Solve Type - Mathematical
An airplane accelerates down a runway at 10 \, m/s^2. It reaches a final velocity of 200 \, m/s until is finally lifts off the ground. Determine the distance traveled before takeoff.
Question 2
Difficulty - Beginner
Solve Type - 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.
Question 3
Difficulty - Beginner
Solve Type - Mathematical
A car decelerates from 25 \, m/s to 5 \, m/s at 10 \, m/s^2. How far does the car travel during this deceleration?
Question 4
Difficulty - Beginner
Solve Type - Mathematical
A car traveling at 20 \, m/s decelerates at a constant rate to a complete stop after traveling 40 \, m.

#### Kinematics in the vertical (y) direction

Question 1
Difficulty - Beginner
Solve Type - 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.
Question 2
Difficulty - Beginner
Solve Type - Mathematical
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.
Question 3
Difficulty - Beginner
Solve Type - Mathematical
A rock is thrown vertically upward with a velocity of 20 \, m/s from the edge of a bridge 42 \, m above a river.

#### Challenge (Bonus) kinematics questions

Question 1
Solve Type - 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?

Question 2
Solve Type - 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.

### 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 a_x and velocity as v_x. 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 \, \frac{m}{s^2}.

### 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

Writer | Coach | Builder | Fremont, CA

## 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

Unit 8 – Fluids

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##### Made By Nerd-Notes.com
KinematicsForces
\Delta x = v_i t + \frac{1}{2} at^2F = ma
v = v_i + atF_g = \frac{G m_1m_2}{r^2}
a = \frac{\Delta v}{\Delta t}f = \mu N
R = \frac{v_i^2 \sin(2\theta)}{g}
Circular MotionEnergy
F_c = \frac{mv^2}{r}KE = \frac{1}{2} mv^2
a_c = \frac{v^2}{r}PE = mgh
KE_i + PE_i = KE_f + PE_f
MomentumTorque and Rotations
p = m v\tau = r \cdot F \cdot \sin(\theta)
J = \Delta pI = \sum mr^2
p_i = p_fL = I \cdot \omega
Simple Harmonic Motion
F = -k x
T = 2\pi \sqrt{\frac{l}{g}}
T = 2\pi \sqrt{\frac{m}{k}}
ConstantDescription
gAcceleration due to gravity, typically 9.8 , \text{m/s}^2 on Earth’s surface
GUniversal Gravitational Constant, 6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2
\mu_k and \mu_sCoefficients of kinetic (\mu_k) and static (\mu_s) friction, dimensionless. Static friction (\mu_s) is usually greater than kinetic friction (\mu_k) as it resists the start of motion.
kSpring constant, in \text{N/m}
M_E = 5.972 \times 10^{24} , \text{kg} Mass of the Earth
M_M = 7.348 \times 10^{22} , \text{kg} Mass of the Moon
M_M = 1.989 \times 10^{30} , \text{kg} Mass of the Sun
VariableSI Unit
s (Displacement)\text{meters (m)}
v (Velocity)\text{meters per second (m/s)}
a (Acceleration)\text{meters per second squared (m/s}^2\text{)}
t (Time)\text{seconds (s)}
m (Mass)\text{kilograms (kg)}
VariableDerived SI Unit
F (Force)\text{newtons (N)}
E, PE, KE (Energy, Potential Energy, Kinetic Energy)\text{joules (J)}
P (Power)\text{watts (W)}
p (Momentum)\text{kilogram meters per second (kgm/s)}
\tau (Torque)\text{newton meters (Nm)}
I (Moment of Inertia)\text{kilogram meter squared (kgm}^2\text{)}
f (Frequency)\text{hertz (Hz)}

General Metric Conversion Chart

Example of using unit analysis: Convert 5 kilometers to millimeters.

1. Start with the given measurement: \text{5 km}

2. Use the conversion factors for kilometers to meters and meters to millimeters: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}

3. Perform the multiplication: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}

4. Simplify to get the final answer: \boxed{5 \times 10^6 \, \text{mm}}

Prefix

Symbol

Power of Ten

Equivalent

Pico-

p

10^{-12}

Nano-

n

10^{-9}

Micro-

µ

10^{-6}

Milli-

m

10^{-3}

Centi-

c

10^{-2}

Deci-

d

10^{-1}

(Base unit)

10^{0}

Deca- or Deka-

da

10^{1}

Hecto-

h

10^{2}

Kilo-

k

10^{3}

Mega-

M

10^{6}

Giga-

G

10^{9}

Tera-

T

10^{12}

1. Some answers may be slightly off by 1% depending on rounding, etc.
2. Answers will use different values of gravity. Some answers use 9.81 m/s2, and other 10 m/s2 for calculations.
3. Variables are sometimes written differently from class to class. For example, sometime initial velocity v_i is written as u ; sometimes \Delta x is written as s .
4. Bookmark questions that you can’t solve so you can come back to them later.
5. Always get help if you can’t figure out a problem. The sooner you can get it cleared up the better chances of you not getting it wrong on a test!

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