1

Equations of Motion

Graphing Equations of Motion

Interpreting Graphs

2

It’s all Relative (Newton)

• In Newtonian Physics all motion is relative. Meaning that it has a reference point.

3

Scalars vs. Vectors • Scalars: One dimension - size. Look at your equation sheet HAS ONLY SIZE

– Mass: Kg – Distance – a measure between any two points two - feet, meters, miles, km – Speed: distance/time– miles per hour, meters per second – Time: seconds, minutes, years – The are always POSITIVE

• Vectors: Two dimensions: SIZE AND DIRECTION – Weight: Newton = Kg m/s2

– Displacement: where the object is in relation to where it started - feet, meters, miles, km in a direction

– Velocity: displacement/time – miles per hour in a direction, meters per second in a direction

– Acceleration: change in velocity in a period of time – m/s2

– They are “Signed” numbers. Can have positive value or negative value

+ y

- y

+ x - x

4

• Scalars can be completely described by one dimension: SIZE – Complete the units for these:

• Mass • Time • Length • Distance • Speed

• Vectors have two dimensions – SIZE AND DIRECTION:

– Complete the Units for these: • Weight • Displacement • Velocity • Acceleration • Force • Torque

Scalars vs. Vectors

2.1 Distance and Speed: Scalar Quantities

• speed – the rate at which distance is travelled • Speed is a scalar quantity • SI units: m/s • average speed – distance divided by time ave. sp. = d t • instantaneous speed – how fast something is moving at a particular instant in time example: your car speedometer example: You walk to Sunoco, 0.5 km away, then walk straight back. The whole trip took 20 min. What was your average speed? 1km /.33 hr =

5

2.2 One-Dimensional Displacement and Velocity: Vector Quantities

displacement – how far and in what direction • displacement is a vector

vector quantity – has magnitude AND direction • represented by arrows • the length of the arrow represents the magnitude

example: A Derry HS student walks from the Office to the Library, 16 m. - Set up a Cartesian coordinate system with the student at the origin. - Orient the motion along one of the axes. initial position x1 = 0.0 m x1 x2 final position x2 = 16.0 m x 0.0 5.0 10.0 15.0 (meters) Δ x = x2 – x1 where is Δ x the change in position, or

displacement (Bold means it is a vector.)

Δ x = x2 – x1 = 16.0 m – 0.0 m

Δ x = 6

2.2 One-Dimensional Displacement and Velocity: Vector Quantities

example: A student walks 12.0 m from the Library to the Guidance. What is her displacement? initial position x2 = 16.0 m x1 Office x3 Guidance x2 Library final position x3 = 4.0 m x 0.0 5.0 10.0 15.0 (meters) Δ x = x3 – x2

Δ x = x3 – x2 = 4.0 m – 16.0 m

Δ x =

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2.2 One-Dimensional Displacement and Velocity: Vector Quantities

velocity – how fast something is moving and in what direction • speed is a scalar; velocity is a vector • SI units are m/s average velocity = displacement time v = Δ x = x – xo or v = x or x = v t Δ t t – to t

instantaneous velocity – how fast something is moving, and in what

direction at a particular instant in time

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Which Velocity is It?

There are two types of velocity that we encounter in our everyday lives. Instantaneous velocity refers to how fast something is moving at a particular point in time, while average velocity refers to the average speed something travels over a given period of time.

For each use of velocity described below, identify whether it is instantaneous velocity or average velocity. 1. The speedometer on your car indicates you are going 65 mph. __________ 2. A race-car driver was listed as driving 120 mph for the entire __________ race. 3. A freely falling object has a speed of 19.6 m/s after 2 seconds of fall in a vacuum. __________ 4. The speed limit sign says 45 mph. __________

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2.3 Acceleration

acceleration – the time rate of change of velocity • acceleration is a vector quantity; SI units are m/s2

average acceleration = change in velocity change in time a = Δ v = v – vo or a = v – vo

Δ t t – to t

instantaneous acceleration – the acceleration at a particular instant in

time

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example: A Derry track team member does a wind sprint from the Library

to the Office and back. His team mate times him at 12.30 s. What was his

average speed? What was his average velocity? Office = 0.0 m x1 Office x2 Library Library = 16.0 m x 0.0 5.0 10.0 15.0 (meters) ave. sp. = d = 32.0 m = 2.60 m/s t 12.30 s v = Δ x = 16.0 m – 16.0 m = 0.0 m/s t 12.30 s

2.2 One-Dimensional Displacement and Velocity: Vector Quantities

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12

Time taken to fall any distance under “g” t = √2dy

g

Time

Points in – y direction - toward center of the

earth

- 9.82 m/s2 -g Acceleration due to gravity

Average velocity used m or miles, etc. df = vf + vi t

2

Displacement during uniform

acceleration

When time is NOT given. m

s

vf2

= vi2 + 2 ad Final velocity if time is NOT given

1.vf = final velocity

2.vi = initial velocity

m

s

vf = vi + at

vy = vi y - gt

Final Velocity after uniform

acceleration

1.How far has the object gone and in what

direction

2.di = any distance already accumulated

3.vit = distance already covered due to being in

motion at a constant velocity

4.½ at2 distance accumulated due to

acceleration

m or miles, etc. d = vt

dx = vix t

df = di + vit + ½ at2

dy = diy + viyt - ½ gt2

Distance x

Scalar

Distance x (Range)

Displacement x

Vector

Projectiles /Free Fall

1.Vector

2.Change in Velocity

Time during which velocity changes

m

s2

a = v

t

a = vf – vi

Δ t

Acceleration - Vector

1.How far

2.How long

3.What direction

4.DISPLACEMENT over time

m mi etc.

s hr

v = d

t

Velocity - Vector

1.How far

2.How long

3.DISTANCE over time

m mi etc.

s hr

v = d

t

Speed - Scalar

Remarks Units Equation Name

d = di + vi(t) + 1/2 a(t)2 d = di + v0(t)

t

d

0

v

0 t

d d

d = di

t

v

t v

0

t

d

t

d

0

1 2 3

6 5 4

Slow, Rightward(+) Constant Velocity

Fast, Rightward(+) Constant Velocity

Position vs. Time Graphs

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15

16

Position vs. Time Graphs

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2.3 Acceleration

Velocity vs. Time Graphs

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How to read these graphs: +x or –x is the LOCATION of the velocity. + or – sign for ACCELERATION is the slope

Velocity vs. Time Graphs

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Object moving slower in –x

- v and +a (slope is +) Object moving slower in +x direction

-v and – a (Slope is -)

+x

-x

+x

-x

+x

-x

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Velocity vs. Time Graphs, Continued

Object moving Faster in +x

+v and +a (slope is +) Object moving slower in +x direction

+v and – a (Slope is -)

+x -x

+x -x

2.3 Acceleration

Time

Acc

ele

rati

on

-

0

+

Time A

cce

lera

tio

n

-

0

+

Time

Acc

ele

rati

on

-

0

+

Time

Acc

ele

rati

on

-

0

+

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These are graphs of the acceleration of the graphs on the former pages

Acceleration refers to any change in an object’s velocity. Velocity not only refers to an object’s speed but also its direction. The direction of an object’s acceleration is the same as the direction of the force causing it.

***************************************************************

Complete the table below by drawing arrows to indicate the directions of the objects’ velocity and acceleration.

Description of Motion Direction of Velocity

Direction of Acceleration

A ball is dropped from a ladder.

A car is moving to the right when the driver applies the brakes to slow down.

A ball tied to a string and being swung clockwise is at the top of its circular path.

A sled is pushed to the left causing it to speed up.

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2.3 Acceleration

Check for Understanding:

The object whose motion is represented by this graph is ... True or false for each?

a) moving in the positive direction.

b) moving with a constant velocity.

c) moving with a negative velocity.

d) slowing down.

e) changing directions.

f) speeding up.

g) moving with a positive acceleration.

h) moving with a constant acceleration.

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24

Constant

Velocity

v0

t

v = 0 v

t

v0

t t

d =

v0(t)

a = 1/2 bh

b= t

h = at (v =

at)

a= 1/2 at2

d =

v0(t)

Acceleration

1 3 2

Constant Velocity

v

0

t

v

0

t

a = 1/2 bh

b= t

h = at (v = at)

a= 1/2 at2

d = vt

Acceleration

Area under the curve of a Velocity versus time graph represents Distance covered in that period of time

1. Constant velocity

2. The area UNDER the slope:

a. Base = time

b. Height = velocity

Distance = velocity x time therefore, the area under the curve of a velocity versus time graph gives you the distance

covered in that time period. d = vt

1.Acceleration graph

a. Slope represents Acceleration v = at

b. Makes a triangle

c. Equation for area of a triangle = ½ bh

d. If the base of the triangle is t and the height is at then the area of this triangle is ½ at2

Since the triangle does not go through the zero we have to add the rectangle to come up with the total area

a = 0

+v, +p

0v, +p

Forward

Backing up -v, +p

0v, +p

0v, +p

0v, -p

-v, +p Backing up

-v, -p Backing up

+v, -p Forward

+p

-p

0v

0v

0v

0v

+p

-p