The 5 Kinematic Equations

The average speed is equal to the total distance traveled divided by the total time of travel

d v = ----t

During an acceleration the average speed is equal to midpoint between the initial and final velocity

vf + vi

2 v =

Find the average speed in the first 4 seconds Find the average speed in the

first 2 seconds

V(2) = ?

V(1) = ?

We can rearrange these equations to create other useful equations..

Goal: Let’s derive an equation for the distance traveled by an accelerating object

v = ----t

d vf + vi

2 =

Now solve the right side for d: d = v t

Substitute v = (vf + vi ) / 2 d = (vf + vi ) t

2Now lets introduce

a = v/t = (Vf – Vi)/t

So t = (Vf – Vi)/a

which we can substitute above…..

Now we can expand out the parenthesis and derive equation #5 from this………..

d = (vf + vi ) (Vf – Vi)/a

2

d = (vf + vi ) (Vf – Vi)

2 a

Vf2 = Vi

2 + 2 a d This is a handy equation used to solve

problems like this:

A car accelerates from a speed of 20 m/s to 30 m/s at an acceleration of 2.0 m/s2.

How far will the car travel during this acceleration?

How fast will an object dropped from rest be moving after 4 seconds?

How far will an object dropped from rest have fallen after 4 seconds?

The rearrangement of equation #2………

a = v/t Solve this for v : v = at = Vf -Vi

Solve this for Vf:

vf = vi + at which is equation #3

Ex: A rocket moving at 1000 m/s accelerates at 20 m/s2 for 10 seconds to achieve a higher orbit. What will its new speed be after it reaches its new orbit?

D

And that’s equation #4: d = vi t + ½ at2

Example: A car traveling at 10 m/s accelerates for 5.0 s at a rate of 2.5 m/s2. How far will it travel during those 5.0 seconds?

Many of his experimental apparatus can be found at the Museo di Storia della Scienza in Florence, Italy

There we have the 5 Kinematic Equations. Galileo didn’t exactly write them algebraically like this, but he set up the ideas for measuring v, a, d and t and showed through reasoning, measurement proportions how v, d , a and t related to one another. Note there is no concept of force yet in Physics…that will come later with Isaac Newton.

d = vi t + ½ at20

d t ½ a t2

Distance vs Displacement5 meters east in 2 seconds

3 meters south in 1 second

The total distance d traveled is

The total displacement d is

Speed = d / t = v

Velocity= d / t = v

Vector mathematics is different than scalar mathematics

Vectors have direction and must be drawn to scale and connected head-to-tail. They add up to a resultant.

Scalars are just numbers and can be added with regular math. They add up to a sum.

HEADTail

Start

Finish

2 miles

4 miles 3 miles

Road runner travels 5 miles before Marvin sends him back. Road runner takes a shortcut back that saves him a mile. If the whole round trip took 1 hour, what is roadrunner’s

a) speed? b) displacement ?