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Kinematic equations

We often we need to describe the motion of an object.

If we can do this without worrying about the cause of that motion, then we’re dealing with kinematics.

Using kinematics in music videos Kinematics-in-music-videos

The hard part: An infinite number of objects, can move in an infinite number of ways. Who’s got time for that?

The easy part: Every moving object follows simple rules, each of which can be expressed with 9th grade algebra!

Motion of an object with constant velocity

Motion of an object that starts with a velocity, and then accelerates

For objects with a constant velocity (accel = 0)

d = v·t

v = Δd /Δt

t = d / v

For objects with a changing velocity (accel not 0)

a = Δv / Δt            definition of accleration

d = ½·at2             Valid when starting from rest: v initial = 0 m/s

d = vit  + ½at2      Valid when starting from any vi (initial speed),
when the starting point is the origin (d = 0)

d = dinitial  +  vit  + ½at2   Valid when starting from any initial speed,

                                                 from starting point dinitial

v =  a·t                                    Valid when starting from rest  v initial = 0 m/s

vf =  vi + a·t                            Valid when starting from any vi (initial speed)

vfinal2  =  vinitial2  + 2a(x – xi)     If we don’t know the distance traveled,

                                                            but we do know the beginning and end

                                                            velocities, then we use this.

v avg  =  (v + vi) / 2                  The is just definition of an average.

How do we solve kinematics problems?


Problem solving strategy

Read the problem. Make a basic sketch of what’s happening.

List the knowns

List the unknowns

Choose the proper kinematic equation.

Then solve for the needed variable

Example: Given a car’s final velocity, time and acceleration, find it’s initial velocity.

Look at the above list of kinematic equations – find one which relates these variables.

Use  vf =  vi + a·t

Algebraically solve for vi

vi = vf – a·t

Always manipulate equations with variables:
don’t put the “knowns” (numbers) in yet.

Only after we have isolated our variable do we put our numbers in:

vi = vf – a·t

vi = 30 m/s – (2 m/s/s)· (3 seconds)

Solve for your result.

vi  =  30 m/s – 6 m/s      =   24 m/s

Is it always this simple? Nope.

The above strategy assumed that there was just one equation that could relate all of our knowns and unknowns. That’s not always the case.  Sometimes we’ll need to solve the problem in several steps, each step might require a different equation.  We’ll see that kind of problem solving more often in physics at the SAT Physics or AP Physics level.

How do we write our answers?

Show how you obtained your answer

use correct units.

realistic number of significant digits
(no matter how many digits appear on the calculator)

Is kinematics sometimes hard? Sure, but remember this:

Neil DeGrasse You Can Always Become Better At It

Something special: The brachistochrone – curve of quickest descent.

Learning standards

2016 Massachusetts Science and Technology/Engineering Standards

HS-PS2-1. Analyze data to support the claim that Newton’s second law of motion is a mathematical model describing change in motion (the acceleration) of objects when acted on by a net force.

HS-PS2-10(MA). Use free-body force diagrams, algebraic expressions, and Newton’s laws of motion to predict changes to velocity and acceleration for an object moving in one dimension in various situations.

A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)
PS2.A Forces and motion. How can one predict an object’s continued motion, changes in motion, or stability?

Massachusetts Science and Technology/Engineering Curriculum Framework (2006)
Introductory Physics. Motion and Forces. Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.

1.1 Compare and contrast vector quantities (e.g., displacement, velocity, acceleration force, linear momentum) and scalar quantities (e.g., distance, speed, energy, mass, work)

1.2 Distinguish between displacement, distance, velocity, speed, and acceleration. Solve problems involving displacement, distance, velocity, speed, and constant acceleration.

1.3 Create and interpret graphs of 1-dimensional motion, such as position vs. time, distance vs. time, speed vs. time, velocity vs. time, and acceleration vs. time where acceleration is constant.

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