Derivation of Kinematic Equations

Where do the equations come from?

Constant velocity

Average velocity- velocity over time

On a graph, Average velocity = the slope of a position vs time graph when an object travels at constant velocity.

v x

t

Displacement when object moves with constant velocity

• Displacement = distance with direction

• The displacement is the area under a velocity

vs time graph (base x height)v

t

x

x v t

Uniform acceleration

This is the equation of the line of the velocity vs time graph when an object is undergoing uniform acceleration.

The slope is the acceleration

The intercept is the initial velocity

v f at v 0v

t

v0

v

t

Displacement when object accelerates from rest

Displacement is still represented by the area under the velocity vs time graph. However,

velocity is constantly changing.

v

t

Displacement when object accelerates from rest

Displacement is still the area under the velocity vs time graph. Use the formula for the area of a triangle.

x 12vt

v

t

v

On a graph…..

displacement = ½ bh

Displacement when object accelerates from rest

From slope of v-t graph

Rearrange to get

Now, substitute for ∆v

a v

t

v a t

x 12a t t

v

t

v

Displacement when object accelerates from rest

Simplify:

Assuming uniform acceleration and a starting time = 0, the equation can be written:

v

t

v

x 12a (t)2

x 12

at 2

Displacement when object accelerates with initial velocity

Break the area up into two parts:

the rectangle representingdisplacement due to initial velocity

v

t

v0

x v 0t

Displacement when object accelerates with initial velocity

Break the area up into two parts:

and the triangle representingdisplacement due to acceleration

x 12a(t)2

v

t

v0

v

Displacement when object accelerates with initial velocity

Sum the two areas:

Or, if starting time = 0, the equation can be written:

x v 0t 12

a(t )2

x v0t 12at 2

v

t

v0

v

Time-independent relationship between ∆x, v and a

Sometimes you are asked to find the final velocity or displacement when the length of time is not given.

To derive this equation, we must start with the definition of average velocity:

v x

t

Time-independent relationship between ∆x, v and a

Another way to express average velocity is:

v x

t

v v f v 0

2

Time-independent relationship between ∆x, v and a

We have defined acceleration as:

This can be rearranged to:

and then expanded to yield :

a v

t

t v

a

t vf v0

a

Time-independent relationship between ∆x, v and a

Now, take the equation for displacement

and make substitutions for average velocity and ∆t

x v t

Time-independent relationship between ∆x, v and a

x v t

v v f v 0

2

t vf v0

a

Time-independent relationship between ∆x, v and a

x v t

x vf v0

2vf v0

a

Time-independent relationship between ∆x, v and a

Simplify

x vf v0

2vf v0

a

x vf

2 v02

2a

Time-independent relationship between ∆x, v and a

Rearrange

x vf

2 v02

2a

2a x v f2 v 0

2

Time-independent relationship between ∆x, v and a

Rearrange again to obtain the more common form:

2a x v f2 v 0

2

vf2 v0

2 2a x

Which equation do I use?

First, decide what model is appropriateIs the object moving at constant

velocity? Or, is it accelerating uniformly?

Next, decide whether it’s easier to use an algebraic or a graphical representation.

Constant velocity

If you are looking for the velocity,use the algebraic form

or find the slope of the graph (actually the same thing)

v x

t

Constant velocity

If you are looking for the displacement, use the algebraic form

or find the area under the curve

x v t

v

t

x

Uniform acceleration

If you want to find the final velocity, use the algebraic form

If you are looking for the acceleration rearrange the equation above

which is the same as finding the slope of a velocity-time graph

vf at v0

a vt

Uniform acceleration

If you want to find the displacement, use the algebraic form eliminate initial velocity if the object

starts from rest

Or, find the area under the curve

x v0t 12at 2

v

t

v0

v

If you don’t know the time…

You can solve for ∆t using one of the earlier equations, and then solve for the desired quantity, or

You can use the equation

rearranging it to suit your needs

vf2 v0

2 2a x

All the equations in one place

constant velocity uniform acceleration

vf2 v0

2 2ax

x v0t 12at 2

t

va

v f at v 0

t

xv

x v t