Lesson 2-1Displacement and Velocity

Displacement Lets say you travel from here to Pittsburgh Many different ways, by boat, by car, by

plane Different methods mean different amounts of

time

Displacement End point is always the same

To describe the results of your motion you need to specify Distance from starting point Direction of travel Direction and distance mean

Displacement is a vector

Back to the Pittsburgh example Displacement is the same no matter what method of travel

or how many stops, starts, or detours

Displacement Displacement is the change in position SI unit is the meter Usually talk about displacement of objects

that move An object at rest has zero displacement

No matter how much time passes, the object will not move

Displacement Displacement is NOT equal to distance

traveled Think of “Something moved around, what is the

shortest distance it could have taken?” Nascar races have zero displacement In football

Offense hopes for positive displacement Defense hopes for negative displacement

Reference Points Coordinate systems are useful to describe

motion Yard markers help on a football field Squares on a chess board

A meter stick is helpful to determine displacement

Reference Points Lets say we have a ball

The ball begins at 15 cm We refer to the starting point as xi

The ball rolls to the 45 cm mark We refer to the ending point as xf

Displacement is found by subtraction Final position – starting position

The Displacement Equation Final position –

starting position Recall Δ means

‘change in’ The displacement

equation is:

x x xf i

Direction of Displacement Displacement may also occur in the vertical

direction A helicopter sits on a heli-pad 30 m above the

ground, it takes off and hovers 200 m above the ground What is the yi? What is the yf? What is the Δy?

yi = 30 m, yf = 200 m, Δy = 170 m

Sign on Displacement Displacement may be positive or negative

From our equation Δx = xf – xi we see Δx is positive if xf > xi

Δx is negative if xf < xi

There is no such thing as a negative distance A –Δx simply tells a direction

Sign on Displacement Coordinate directions

Using ‘right’ as positive and ‘left’ as negative is only by convention That does not mean it is necessarily correct

As long as you remain constant throughout the situation, you may call ‘left’ positive. Thus making ‘right’ negative

Similarly, you may call ‘down’ positive Thus making ‘up’ negative

Displacement Practice

1) xi = 10 cm, xf = 80 cm

2) xi = 3 cm, xf = 12 cm

3) xi = 80 cm, xf = 20 cm

4) xi = 28 cm, xf = 11 cm

70 cm

9 cm

-60 cm

-17 cm

Concept Chall. Pg 41

Velocity Quantity that measures how fast something

moves from one point to another Different than speed, Velocity has direction

Speed is the magnitude part of the velocity vector Velocity has direction and magnitude

Average Velocity To calculate, you must know the time the

object left and arrived Time from initial position to final position Avg. Vel. is displacement divided by total time

vx

t

x x

t tavgf i

f i

Avg. Velocity vs Avg. Speed Main difference

Average Velocity depends on total displacement (direction)

Average speed depends on distance traveled in a specific time interval

Lesson 2-2Acceleration

Acceleration Lets say you are driving at 10 m/s

You approach a stop sign and brake carefully and stop after 6 seconds

Your speed changed from 10 m/s to 0 m/s over that time

Lets say you had to brake suddenly and stopped after 2 seconds

Your speed changed from 10 m/s to 0 m/s over that time

Acceleration What was the main difference between those

two examples? Time

A slow, gradual stop is much more comfortable than a sudden stop

Average Acceleration The quantity that describes the rate of change

of velocity in a given time interval is acceleration

av

t

v v

t tavgf i

f i

Average Acceleration Units of acceleration are length per seconds squared

Analysis:

av

t

m s

sm s s m savg

// / 2

Constant Acceleration As an object moves with constant a, the V

increases by the same amount each interval There is a very specific relationship between

displacement, acceleration, velocity, and time The relationship is used to produce a group of

very important equations

Kinematic Equation #1 Displacement depends

on acceleration, initial velocity and time and

vx

tavg

vv v

avgf i2

x

t

v vf i

2

xv v

tf i

2

x v v tf i 1

2( )Kinematic Equation #1:

Kinematic Equation #2 Final velocity depends

on initial velocity, acceleration and time

av

t

v v

tf i

a t v vf i

v a t vi f

v v a tf i Kinematic Equation #2:

Kinematic Equation #3 We can form another

equation by plugging #2 into #1

x v v ti f 1

2( ) v v a tf i

x v v a t ti i

1

2( ( ))

x v a t ti

1

22( )

x v a t ti( )1

2

Kinematic Equation #3:

x v t a ti ( )1

22

Kinematic Equation #4 So far, all of our Kinematic Equations have

required time interval What if we do not know the time interval We can form one last equation by plugging

equation #1 into #2

Kinematic Equation #4

x v v ti f 1

2( )

LNM

OQP2 2

1

2 x v v ti f( )

2 x v v ti f( )

2

x

v vt

i f( )

v v a tf i

LNMM

OQPPv v a

x

v vf ii f

2( )

LNMM

OQPPv v a

x

v vf ii f

2( )

( )( )v v v v a xf i i f 2

Kinematic Equation #4

( )( )v v v v a xf i i f 2

v v a xf i2 2 2

Kinematic Equation #4: v v a xf i2 2 2

Note: A square root is needed to find the final velocity

Lesson 2-3Falling Objects

Free Fall In a vacuum, with no air, objects will fall at

the same rate Objects will cover the same displacement in the

same amount of time Regardless of mass We cannot demonstrate this because of air

resistance

Gravity Objects in free fall are affected by what?

Gravity A falling ball moves because of gravity

“The force of gravity” Gravity is NOT a force!! Gravity is an acceleration

Gravity as an Acceleration Since acceleration is a vector

Gravity has magnitude and direction Magnitude is -9.81 m/s2 or 32 ft/s2

Direction is toward the center of the Earth Usually straight down

Gravity is denoted as g rather than a Gravity is a special type of acceleration

Always directed down, so the sign should always be negative

-9.81 m/s2 or -32 ft/s2

Path of Free Fall If a ball is thrown up in the air and falls back

down the same path, some interesting things happen At the maximum height, the ball stops

As the ball changes direction, it may seem as V and a are changing

V is constantly changing, a is constant from the beginning a is g throughout

Path of Free Fall At ymax

What is the velocity? 0 m/s

What is the acceleration? g or -9.81 m/s2

Free Fall It may be tough to think of something moving

upward and having a downward acceleration Think of a car stopping at a stop sign

When an object is thrown in the air, it has a +Vi and –a Since the two vectors are opposite each other, the

object is slowing down

Free Fall The velocity decrease until the ball stops and

velocity is 0 It is tough to see the ‘stop’ since it is only for a

split second Even during the stop, a = -9.81 m/s2

What happens after the ball stops at the top of its path?

Free Fall The ball begins to free fall

When the ball begins to move downward It has a negative velocity It has a negative acceleration V and a now in the same direction Ball is speeding up

This is what happens to objects in free fall They fall faster and faster as they head toward

Earth