KINEMATIC EQUATIONS

New equations and how to use them!

KINEMATIC EQUATIONS Kinematic Equations are considered to be

“equations of motion” and are based on the fundamental definitions of average velocity and acceleration:

t

dv

t

vva 12

221 vv

v

OUR VARIABLES There are 5 basic variables that are used in

any motion-related calculation: Initial Velocity = v0 or vi or v1Final Velocity = v or vf or v2Acceleration = aDisplacement = d (sometimes also s )Time = t

Bold face indicates a vector Each of the kinematic equations will use 4

of these 5 variables

Each of the kinematic equations starts with a rearranged version of the equation for average velocity:

And uses substitution, rearranging, and simplifying the equations to get to the end result.

For example…

DERIVING THE EQUATIONS

tvd

KINEMATICS EQUATION #1

Step 1: Step 2: Substitute

equation for Step 3: Rearrange

acceleration equation to solve for t, then substitute

Step 4: Simplify by multiplying fractions

Step 5: Rearrange

tvd

tvv

d

2

21

a

vvvvd 1212

2

v

a

vvd

2

21

22

21

222 vvad

advv 221

22

221 vv

v

→

→a

vvt 12

KINEMATICS EQUATION #2

Step 1:

Step 2: Substitute Step 3: Rearrange

acceleration equation to solve for v, then substitute

Step 4: Simplify Step 5: Distribute the

t through the equation

Step 6: Simplify again

tvd

tvv

d

2

12

212 vv

v

→

atvv 12 tvatv

d

2

)( 11

tatv

d

2

2 1

2

2 21 attv

d

21 2

1attvd

→

SUMMARY OF EQUATIONS

You will NOT be required to memorize these

atvv 12

21 2

1attvd

advv 221

22

The equation of the displacement-time graph is:

The slope of this graph = velocityThe y-intercept of this graph = initial position

(displacement)

HOW DO THESE RELATE TO OUR LABS?

1dvtd

The equation of the velocity-time graph is:

The slope of this graph = accelerationThe y-intercept of this graph = initial velocity

HOW DO THESE RELATE TO OUR LABS?

1vatv

PROBLEM SOLVING STRATEGY

When given problems to solve, you will be expected to “show your work” COMPLETELY!

“Showing work” means that you will be expected to include the following pieces in your full answer (or you will not receive full credit for the problem…)List of variables – include units on this listEquation – in variable form (no numbers plugged in

yet) If necessary, show algebra mid-steps (still no

numbers)Plug in your value(s) for the variablesFinal answer – boxed/circled with appropriate

units and sig figs

A school bus is moving at 25 m/s when the driver steps on the brakes and brings the bus to a stop in 3.0 s. What is the average acceleration of the bus while braking?

v2 =

v1 =

t = a =

PRACTICE PROBLEM #1

25 m/s

0 m/s

3.0 s

?

a = -8.3 m/s2

atvv 12

t

vva

atvv

12

12

s0.3s

m250 sm

a

PRACTICE PROBLEM #2 An airplane starts from rest and

accelerates at a constant 3.00 m/s2 for 30.0 s before leaving the ground.(a) How far did it move?(b) How fast was it going when it took off?

v2 =

v1 =

t = a =d =

0 m/s

?

30.0 s

3.00 m/s2

d = 1350 m?

v2 = 90.0 m/s

21 2

1attvd

2)0.30)(00.3(2

10d

atvv 12

)0.30)(00.3(02 v