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Kinematics CH 3 Describing motion

What is Motion?

any physical movement or change in position or place, relative to a reference point

Reference Point

Movement

Motion diagrams

At rest Moving at a constant speed

Speeding up Slowing down

A series of images of a moving object that records its

position after equal time intervals.

It represents the position, velocity and acceleration

of an object at several different times.

the particle model

What is the motion of the cart in this

diagram?

Slowing down (deceleration)

Change is position

is less/time interval

We can use motion diagrams to

represent 4 concepts in kinematics:

Uniform motion (constant speed)

At rest

Speeding up (acceleration)

Slowing down (deceleration)

Choose the coordinates

Establish where “0” is (the origin)

Establish directions where the values increase

Using a coordinate system

example - football

What origins and direction are used to determine 1st down?

Length of punt?

Position vector

Proportional to the distance of the object from the

origin and points from origin to the location of the

object at a particular time.

0

+

+

When would position be –

but the displacement +?

distance and displacement

Distance refers to how much ground an object

has covered during its motion.

X Axis

Y Axis

Distance = how far an object has moved.

Measured in meters, kilometers (cm or mm)

If each mark represents 10 cm, what is the distance between the girl and the ball? ______

displacement ≠ distance

Displacement is the object's change in

position.

X Axis

Y Axis If the girl walks to the red ball, then walks backwards to the bear, what distance has she traveled? ______

Displacement = the distance of a body's change in position from a starting point. Her final displacement is ______.

©2008 by W.H. Freeman and Company

The distinction between distance and displacement.

Displacement (blue line) is how far the object is from its starting point,

regardless of how it got there.

Distance traveled (dashed line) is measured along the actual path.

As any object moves from one position to another,

the length of the straight line drawn from its initial

position to the object’s final position is called

displacement.

Displacement doesn’t always tell you distance an

object moved.

Considering

displacement

∆x = xf - xi

Math and physics

Physicists use the tools of math to describe measured or

predicted relationships between physical quantities in a

situation. Equation = a compact statement

based on a model of the situation.

Shows how 2 or more variables are thought to be related.

Physics shorthand

∆ “difference” or “change in”

∑ “sum” or “total”

∆x, ∆y change in position

∆t time interval

Distance and Displacement

Scalar - magnitude only

Vector - magnitude AND direction

Distance is the length of the actual path taken by an object. Consider travel from point A to point B in diagram below:

Distance s is a scalar

quantity:

Contains magnitude only

and consists of a number

and a unit.

ex: 20 m, 40 mi/h, 10 gal

A

B s = 20 m

Displacement is the straight-line separation of

two points in a specified direction.

A vector quantity:

contains magnitude AND

direction.

A

B D = 12 m, 20o

q

ex: 12 m, 300; 8 km/h, N

In th e d ia gra m below, th e or igin , or in it ia l pos it ion (d o) is a t 0 .0 m . Th e

fin a l pos it ion (d 1) is a t 50 m . Th e d is ta n ce t ra veled from th e or igin is 50

m , bu t th e d is p la cem en t is 50 m to th e r igh t , or ca n be d ra wn s im ila r with

a green lin e – vec t or .

How are distance and displacement related to

motion?

Motion of an object is BOTH

a scalar quantity (time)

And

Vector quantity (displacement)

xi

initial position

xf

final position

Displacement: ∆x = xf – xi

displacement = distance and direction between

2 positions = change in position =

final position – initial position

• If displacement is positive, the object

moves to the right.

• If the displacement is negative, the object

moves to the left.

Defining the reference point and direction

The values of xi and xf are determined by their positions on

the axis.

While the choice of a reference point for the coordinate

system is arbitrary, once chosen, the same point must

be used throughout the problem.

Signs of Displacement

In ph ys ics , th e m ovem en t from th e or igin is th ou gh t of a s

pos it ive or n ega t ive. In ea ch ca s e, a n or igin , s ta r t in g p la ce,

or referen ce poin t n eeds to be es ta b lis h ed . Th en , it m u s t be

decided wh ich d irect ion s a re con s idered pos it ive a n d wh ich is

con s idered n ega t ive. On ce a grou p a grees on th a t , th en you

ca n deter m in e d is p la cem en t vectors . For exa m ple, if we

a s s u m e th a t u p is pos it ive, th en Mt. Elin or, wou ld h a ve a n

eleva t ion d is p la cem en t vector

of +2 ,400 ft , ba s ed off of th e or igin

of s ea level. On th e oth er h a n d ,

Dea th Va lley, Ca lifor n ia is below

s ea level by 120 ft , s o its

d is p la cem en t wou ld be -120 ft

com pa red to s ea level.

©2008 by W.H. Freeman and Company

Left:

Displacement is positive.

Right:

Displacement is negative.

The displacement is written:

The Signs of Displacement • Displacement is positive (+) or negative

(-) based on LOCATION.

2 m

-1 m

-2 m

The displacement is the

y-coordinate. Whether

motion is up or down,

+ or - is based on

LOCATION.

Examples:

The direction of motion does not matter!

• For motion along x or y axis, the displacement is determined by the x or y coordinate of its final position.

• Consider a car that travels 8 m, E then 12 m, W.

Net displacement “D” is from the

reference point to the final position:

What is the distance

traveled? 20 m !! 12 m,W

D

D = 4 m, W x

8 m,E

x = +8 x = -4

What is this

measuring?

• Would it be good to measure MPG (miles per gallon) in position, displacement or distance?

•Distance

•Displacement

(Not the speedometer sillies)

Displacement and Distance

• A person starts at the 5.o m mark. They walk

to the 12m mark.

– What is their distance travelled?

– What is their displacement?

• They leave the 12m mark and walk to the 1m

mark.

– What is their distance travelled?

– What is their displacement?

• What is the total distance travelled?

• What is the total displacement for the motion?

• What is the distance

travelled for the first three

seconds?

• When is the object not

moving?

• What is the final speed of

the object at t=3 seconds?

• Rank speed at t=0,1,3

• What is the average velocity

for the entire motion?

What about at angles?

• You drive 3 miles east and then 4 miles

north. What is your:

– Why are you not drawing a picture people?

– Distance travelled?

– Displacement?

Strategies in Solving Problems

1. Picture the problem (See it…)

2. What am I answering?

3. What is important? (variables)

4. What will get me there?

5. Solve it.

• This year, the most important thing in the course is getting stronger at the top four. It’s the hardest thing, but transfers to everything in life.

In this motion diagram the length of the arrow indicates

the change in position of the object, or its _____.

Check it:

a. Displacement

b. Magnitude

c. Position

d. Resultant

In this image, 7 cm is a _____.

a. Vector

b. speed

c. Scalar

d. interval

VELOCITY, SPEED,

AND

ACCELERATION

The Cheetah: A cat that is built for speed. Its strength and

agility allow it to sustain a top speed of over 100 km/h.

Such speeds can only be maintained for about ten seconds.

Definition of Speed • Speed is the distance traveled per unit of time

(a scalar quantity).

s = = d

t

20 m

4 s

s = 5 m/s

Not direction dependent!

A

B s = 20 m

Time t = 4 s

Speed (meters/second) = distance (in meters) time (sec)

s = d t

Speed equation

velocity

A quantity that measures how fast

an object moves from one point to

another in a certain direction.

Definition of Velocity • Velocity is the displacement per unit of

time. (A vector quantity.)

v = 3 m/s at 200 N of E

Direction required!

A

B d = 20 m

Time t = 4 s

12 m

4 s

Dv

t

D=12 m

20o

Velocity

Velocity is a measure of the speed of an object AND the direction it is moving in space. On the escalator, passengers are moving at the same constant speed, but they are moving in different directions. Velocity can change even if speed is remaining constant (you just change direction)

Velocity is defined as a vector quantity that tells the

ratio of the displacement change to the time

change,

or

how fast an object is going and in what direction.

Speed, on the other hand, is merely the magnitude

of the velocity, or how fast an object is moving.

speed and velocity are NOT the same

Velocity – the rate at which an object changes its

position (has direction)

Speed – is the magnitude of velocity (how fast an object is

moving)

In other words: the average Velocity depends

on total displacement

BUT

the average Speed = total distance traveled

time interval

1) Ave rage s pe e d – a vera ge of a ll you r s peeds over th e wh ole t r ip . For exa m ple, on a t r ip to Flor ida , (910 m iles ) th e t ra veler took 13 h ou rs to

get th ere. Th e a vera ge s peed wou ld be 70 m ph .

Th is does n ’t m ea n th a t th e ca r wa s goin g exa ct ly 70 m i/ h r th e en t ire

t im e. Som etim es th e ca r wa s goin g fa s ter, a n d oth ers s lower.

2) Con s t an t Spe e d - t ra velin g a t th e s a m e ra te for a lon g per iod of

t im e. Con s ta n t s peed is h a vin g th e cru is e con trol on in th e ca r. Th e ca r

m a in ta in s th e s a m e s peed th e en t ire t im e you a re clock in g it .

3) In s t an t an e ous s pe e d - ra te a t wh ich a n ob ject is t ra velin g a t a

cer ta in m om en t . Th is is you r s peedom eter in you r ca r. It tells you h ow

fa s t th e ca r is goin g a t th e t im e you look a t it .

Th ere a re th ree types of s peed :

©2008 by W.H. Freeman and Company

Average Speed & Velocity

Velocity includes directional information:

t

x

tt

xxv

12

12

Average velocity = change in position

change in time

= displacement

time interval

V avg = ∆x = xf – xi

∆t tf - ti

V = ∆d = d1 – d0

∆t t1 – t0

• Units are meters/second (m/s)

• Can be positive or negative depending on

direction moved.

v

d

The Signs of Velocity

First choose + direction; then

v is positive if motion is with

that direction, and negative if

it is against that direction.

Velocity is positive (+) or negative (-)

based on direction of motion.

- +

- +

+

example problem

During a race, Carla covers 650 m in 125 s

running east on a straight road. Find Carla’s

average velocity.

V = ∆d = 650 m = 5.2 m/s

∆t 125 s

How long will it take her to run 5 km?

5.2 m/s = .0052 km x 3600 s = 18.7 km

s hr hr

18.7 km = 5 km = .267 hr

hr x hr

Try one

• Heather and Matthew walk eastward with a speed of .98 m/s. If it takes them 34 min to walk to the store, how far have they walked?

• Knowns? What do you know? Write it down.

• Speed = .98 m/s, time = 34 minutes (2040 sec)

• Unknown? What do you want to know?

• How far? Distance = ?

• Equation? Write the equation you’ll use.

• Speed = distance / time

• Work the problem.

• .98 m/s = distance / 2040 sec; d = 2000 meters

Example. A runner runs 200 m, east, then changes

direction and runs 300 m, west. If the entire trip takes

60 s, what is the average speed and what is the

average velocity?

Recall that average speed is

a function only of total

distance and total time:

Total distance: s = 200 m + 300 m = 500 m

500 m

60 s

total pathAverage speed

time Avg. speed

8.33 m/s

Direction does not matter!

start

s1 = 200 m s2 = 300 m

Example 1 (Cont.) Now we find the average

velocity, which is the net displacement divided by

time. In this case, the direction matters.

xo = 0

t = 60 s

x1= +200 m xf = -100 m 0fx xv

t

x0 = 0 m; xf = -100 m

100 m 01.67 m/s

60 sv

Direction of final displacement

is to the left as shown.

Average velocity: 1.67 m/s, Westv

Note: Average velocity is directed to the west.

Example 2. A sky diver jumps and falls for 600 m

in 14 s. After chute opens, he falls another 400 m

in 150 s. What is average speed for entire fall?

625 m

356 m

14 s

142 s

A

B

600 m + 400 m

14 s + 150 s

A B

A B

x xv

t t

1000 m

164 sv 6.10 m/sv

Average speed is a function only

of total distance traveled and the

total time required.

Total distance/ total time:

Walking Trip Tracking speed and velocity

checking for understanding

An Indianapolis 500 car races around the track

at 225 mph. At the end of the race (500 miles),

what was its average velocity?

for example

A book gets pushed around the

perimeter of a table with

dimensions 1.75 m X 2.25 m. It

completes this motion in 23 s.

What is its average velocity?

What is its average speed?

another example

Car A travels from New York to

Miami at a speed of 25 m/s.

Car B travels from New York to

Canada at a speed of 25 m/s.

Are their velocities equal?

Problems

• You run down the road 500m. It takes you 32sec to complete the task.

• What is your:

– average speed?

– average velocity?

– displacement?

– distance?

• You run around a circular track (radius of 300m) in 32 sec

• What is your:

– average speed?

– average velocity?

– displacement?

– distance?

• What if you went half way?

• Will your average speed ever be zero?

one more for good measure

You travel on a straight highway from your house to visit

your friend 370 km (230 mi) to the west. You leave your

house at 10 am and arrive at 3 pm.

However, after you left your house, you realized you

forgot your toothbrush. You were only 15 km down the

road so you went back and got it.

Half way to your friend’s house, you took a short 5 km

side road to grab a burger at your favorite burger place.

What was your average velocity for this trip? What was your average speed for this trip?

Apply it

Could we determine the final position of

an object from its average velocity?

Show the equation you would derive to

solve this type of problem.

v = Δd vΔt = Δd Δd = df – di Δt so: df – di = vΔt df = vΔt

Calculating Velocity and Speed

acceleration The rate of changing velocity

Think about this....

What are three ways to change the velocity of a car?

Accelerate

Decelerate

Change direction

Slow your car to a stop at a stop sign. Slow from 9 m/s

to 0.0 m/s in 5 s.

Slam on the breaks to stop. Slow from 9 m/s to 0.0 m/s

in 1.5 s.

acceleration

The rate at which an object changes

its velocity in a given time.

• Has velocity so must include a direction.

• Any time an object’s velocity is changing,

the object has an acceleration.

Definition of Acceleration

An acceleration is the change in velocity per

unit of time. (A vector quantity.)

A change in velocity requires the application

of a push or pull (force).

A formal treatment of force and acceleration will be given later. For now, you should

know that:

• The direction of

acceleration is same as

direction of force.

• The acceleration is

proportional to the

magnitude of the force.

Change in velocity is often written as a = ∆v

∆t

If a car moves at a constant

velocity, then its

acceleration is zero

acceleration = change in velocity

change in time

acceleration (m/s2) = (vf) - (vi) time

©2008 by W.H. Freeman and Company

Acceleration

Acceleration is the rate of change of velocity.

t

v

tt

vva

12

12

©2008 by W.H. Freeman and Company

Acceleration

Acceleration is a vector, although in one-dimensional

motion we only need the sign.

The previous image shows positive acceleration; here is

negative acceleration:

calculating average acceleration

average acceleration = change in velocity

change in time

a = ∆ v units - v = (m/s) = m X 1 = m

∆ t t s s s s2

Constant Acceleration

Acceleration:

Setting to = 0 and solving for v, we have:

Final velocity = initial velocity + change in velocity

0fv v at

0

0

f

avg

f

v vva

t t t

try solving

Find the average velocity for the opening problem.

Slow your car to a stop at a stop sign. Slow from 9 m/s

to 0.0 m/s in 5 s.

Slam on the breaks to stop. Slow from 9 m/s to 0.0 m/s

in 1.5 s.

a = ∆ v - 9 m/s = - 1.8 m /s2 vf - vi

∆ t 5 s tf - ti

-9 m/s = - 6.0 m /s2

1.5 s

©2008 by W.H. Freeman and Company

Acceleration

There is a difference between negative acceleration and

deceleration:

Negative acceleration is acceleration in the negative direction as

defined by the coordinate system.

Deceleration occurs when the acceleration is opposite in direction to

the velocity.

- the sign of the velocity and the acceleration is the same if

the object is speeding up and that

- the sign of the velocity and the acceleration is the opposite

if the object is slowing down.

Review of Symbols and Units

• Displacement (x, xo); meters (m)

• Velocity (v, vo); meters per second (m/s)

• Acceleration (a); meters per s2 (m/s2)

• Time (t); seconds (s)

Review sign convention for each symbol

One last problem:

Will and grace enter a race. Both run at the same rate. Both

walk at the same rate. Will runs half the distance and walks

half the distance. Grace runs half the time and walks half

the time. Who wins?

Grace wins – she runs a longer distance. When she runs half

the time she covers more distance than when she walks half

the distance. So we know the she runs for more than half the

distance.

• Practice Problems

• pg. 59, 3.3 Section Review

• pg. 61, Chapter Review, problems 17-21

• Kinematics practice problems worksheet