PhysicsPhysicsChapter 5Chapter 5

Position-Time GraphPosition-Time Graph Time is always on the x axisTime is always on the x axis The slope is speed or velocityThe slope is speed or velocity

Time (s)

Pos

ition

(m

)Slope = Δ y

Δ x

Velocity-Time GraphVelocity-Time Graph Time is always on the x axisTime is always on the x axis The slope is accelerationThe slope is acceleration Area under the curve is positionArea under the curve is position

Slope = Δ y Δ x

time (s)

Vel

ocity

(m

/s)

Area under velocity time graph is positionArea under velocity time graph is position

Time (s)

Vel

ocity

(m

/s)

Area = ½ b * hArea = ½ b * h

For this triangleFor this triangle

A = ½ (velocity) (time)A = ½ (velocity) (time)

Acceleration -Time GraphAcceleration -Time Graph Time is always on the x axisTime is always on the x axis Area under the curve is velocityArea under the curve is velocity

time (s)

Acc

eler

atio

n (m

/s/s

)Slope = Δ y

Δ x

Area under Area under acceleration time graph is velocityacceleration time graph is velocity

Time (s)

Acc

eler

atio

n (m

/s/s

)

Area = ½ b * hArea = ½ b * h

For this triangleFor this triangle

A = ½ (acceleration) (time)A = ½ (acceleration) (time)

Acceleration is often Acceleration is often graphed like thisgraphed like this

time (s)

Acc

eler

atio

n (m

/s/s

)

l+

Which makes area Which makes area under the curve …under the curve …

time (s)

Acc

eler

atio

n (m

/s/s

)

l+

Area = b * hArea = b * h

For this For this

A = (acc) (time)A = (acc) (time)

Looking at graphsLooking at graphs Average uses slope of the chordAverage uses slope of the chord Instantaneous uses slope of the Instantaneous uses slope of the

tangenttangent If slope of the chord = slope of If slope of the chord = slope of

the tangent line then average = the tangent line then average = instantaneousinstantaneous

AverageAverageVelocityVelocity

01

01

tt

dd

t

dv

101 tvdd

Which leads to a Kinematic Equation

01

01

tt

ddv

101 tvdd

0101 ddttv

011 ddtv Let time at 0 be 0Let time at 0 be 0

oror

vtdd 0

Position with Position with Constant VelocityConstant Velocity

AverageAverageaccelerationacceleration

01

01

tt

vv

t

va

atvv 0

Which leads to another Kinematic Equation

01

01

tt

vva

atvv 0

0101 vvtta

011 vvta oror

Let time at 0 be 0Let time at 0 be 0

Final position withFinal position withConstant accelerationConstant acceleration

tvvdd 00 21

tvvd 021

Time (s)

Vel

ocity

(m

/s)

d = vd = v00 * t * t

vv00

vv

tt

d = ½ (v – vd = ½ (v – v00) * t) * t

oror

d = ½ vt – ½vd = ½ vt – ½v00tt

Add them together Add them together & you get& you get

If the initial position is not If the initial position is not zero, then add dzero, then add d00 to the to the

total distancetotal distance

tvvdd 00 21

Final position withFinal position withConstant accelerationConstant acceleration

200 2

1 attvdd

200 2

1 attvdd

If v is not known, If v is not known, substitute the substitute the

following equation following equation in for vin for v

tvvdd 00 21

atvv 0

tatvvdd 000 21 This leads This leads

to…to…

Final velocity withFinal velocity withConstant accelerationConstant acceleration

020

2 2 ddavv

advv 220

2 Or simplyOr simply

Final velocity withFinal velocity withConstant accelerationConstant acceleration

020

2 2 ddavv

advv 220

2 Or simplyOr simply

To solve this To solve this equation, note equation, note that it does not that it does not include time.include time. tvvdd 00 2

1

020

2 2 ddavv

atvv 0

a

vvvvdd 0

00 21

a

vvt 0

Solve for tSolve for t

Sub t into:Sub t into:

a

vvvvdd

200

0

20

202 vvdda

Kinematic EquationsKinematic Equations

vtdd 0 atvv 0

tvvdd 00 21

200 2

1 attvdd

020

2 2 ddavv