Introduction to

Kinematics

Motion, Forces and Energy

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Mechanics: The study of motion

Kinematics The description of how things move 1-D and 2-D motion

Dynamics The study of the forces that cause motion Newton’s Laws Energy Momentum

Remember:

Before we can study something, scientists need to define

what it is they are studying. They determine what belongs

in the study boundaries – the system – all processes,

forces, and measurable properties.

Anything outside the system’s boundaries – the

surroundings.

The surroundings can affect the system – and these

interactions are studied as well.

1. Sketch the problem

Remember: problem solving method

3. Check it! 3. Calculate your Answer

2. Choose the formula(s)

Kinematics

What is Motion?

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

Referenceo

int

Movement

understanding vocabulary

A stationary landmark or

location from which we start or

make our observations.

Reference point

Choose the point of reference or reference frame that

is easiest for you to use in solving the problem or

answering the question.

Make all measurements from that point of reference or

reference frame.

understanding vocabulary

An unchanging rate of motion.

Moving at uniform speed.

understanding vocabulary

The rate of change of speed

during motion.

Choose the coordinates

Establish where “0” is (the origin)

Establish directions where the values increase

the origin (value of 0) is placed at the reference point.

understanding vocabulary

Imaginary line marked off in measured units

Using a coordinate system

example - football

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

Length of punt?

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.

0 1 2 5 3 4

the particle model

What is the motion of the cart in this

diagram?

Slowing down (deceleration)

Change is position is less

per each time interval

time in seconds

We can use motion diagrams to

represent 4 concepts in kinematics:

Uniform motion (constant speed)

At rest

Speeding up (acceleration)

Slowing down (deceleration)

Demonstrate uniform and

accelerated motion

Design it, show it, use evidence to support it.

Write it up……..

1. Name, date, “Physical Science”, Title

2. Problem: How can I use a car & track to

demonstrate uniform motion AND accelerated

motion?

3. Diagram and label your set up

4. Collect data in a data table – multiple trials!

5. Graph your results

6. Conclusion: At least one paragraph.

Don’t just say your proved your

hypothesis – DESCRIBE HOW the

result answers the questions.

Collaborate to collect and analyze your data – but

everyone completes their own written lab report. Use

your own words – copying is not collaborating.

SCALARS AND VECTORS

Measuring motion in 1 and 2 dimensions

understanding vocabulary

A measurable quantity with

appropriate units. Can be positive,

negative or zero.

has magnitude

examples: temperature, volume, speed, mass, distance etc.

understanding vocabulary

A measurable quantity that also

includes direction. Can be positive or

negative.

has magnitude

AND

direction

examples:

try some….

a. 5 m See Answer

b. 30 m/sec, East See Answer

c. 5 mi., North See Answer

d. 20 degrees Celsius See Answer

e. 256 bytes See Answer

f. 4000 Calories

Scalar or Vector?

scalar

scalar scalar

scalar

vector

vector

Position vector

Proportional to the distance of the object from the

origin and points from the origin to the location of the

object at a particular time.

0

+

+ 1 2 4 5 3

meters

What is smiley’s position vector?

+ 3 m

Or - 3 m, right

origin

Describe the position of

the house relative to

the bicycle using only

scalars. 4 km

Describe the position of

the house relative to the

bicycle using vectors.

4 km, N

Is speed a scalar or

vector?

1. What is the bike’s speed

after 1 hour? 25 mi/hr

2. Does the bike speed up or

slow down at B? How do you

know? Speeds up, slope

increases

3. What is the speed at B?

B

C

50 mi/hr

4. What is the speed at C?

0, it is stopped

Distance, Direction and

Displacement How far – overall?

remember

“Motion occurs when an object’s position

changes during an interval of time.”

“One dimensional motion can be represented by

position along a single coordinate axis.”

understanding vocabulary

The length of a time span.

initial time - ti

final time - tf

tf - ti represents the time interval

use the formula: Δt = tf - ti

distance vs displacement

Distance is a scalar quantity. It is always positive.

It refers to how much ground an object has covered

during its motion. There is no direction associated with it.

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

Distance d 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 d = 20 m

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.

distance vs displacement

Displacement is a vector quantity.

It is the object's overall change in position. It can be

positive or negative depending on your initial point of

reference.

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.

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 the diagram below, the origin, or initial position (do) is at 0.0 m. The

final position (d1) is at 50 m. The distance traveled from the origin is 50

m, but the displacement is 50 m to the right, or can be drawn similar

with a green line – vector.

We use arrows to show the direction from

the origin.

• 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

X 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 ______.

0 6 10 -2

remember:

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.

∆x = xf - xi

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

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

If displacement is the distance and direction

between 2 positions = change in position =

final position – initial position

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 physics, the movement from the origin is thought of as positive or

negative. In each case, an origin, starting place, or reference point

needs to be established. Then, it must be decided which directions

are considered positive and which is considered negative. Once a

group agrees on that, then you can determine displacement vectors.

For example, if we assume that up is positive, then Mt. Elinor, would

have an elevation displacement vector

of +2,400 ft, based off of the origin

of sea level. On the other hand,

Death Valley, California is below

sea level by 120 ft, so its

displacement would be -120 ft

compared to sea level.

Left:

Displacement is positive.

Right:

Displacement is negative.

displacement is written as:

When would position be – but

the displacement +?

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

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

What is the car’s

displacement?

What is the distance

traveled? 20 m !!

12 m,W

D D = 4 m, W x

8 m,E

x = +8 x = -4

try a problem

• A person starts at the 5.0 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 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?

A

C

B 3 mi

4 mi Distance: AB + BC

3 + 4 = 7 mi

Displacement: measure of the hypotenuse

(AB)2 + (BC)2 = (AC)2

(3)2 + (4)2 = x2 9 + 16 = √25

= 5 mi., NE

Try some on your own

Graphing the Relationship (function)

Distance vs time graphs

We can represent motion

graphically in 1 D using a

motion dot diagram on the

X-axis.

We can represent distance

and displacement either in

1 D or in 2 D using

algebra and vectors.

• You drive 3 miles east and then 4 miles north.

What is your:

– Distance travelled?

– Displacement?

A

C

B 3 mi

4 mi

Distance: AB + BC

3 + 4 = 7 mi

Displacement: measure of the hypotenuse

(AB)2 + (BC)2 = (AC)2

(3)2 + (4)2 = x2 9 + 16 = √25

= 5 mi., NE

What if we want to show the

relationship between the motion and the

time it takes for that motion?

Distance vs time – graphing a linear function

When you create a ratio representing the

distance (or displacement) of an object and the

time it takes for that motion to occur, you are

creating a rate (a derived unit).

Ex: you travel to your house which is 5 miles away

in 30 min. at a constant rate of speed.

Ex: you travel to your house which is 5 miles away

in 30 min. at a constant rate of speed.

What is your rate of speed for this journey?

5 miles X 60 min. = 10 mi./hr

30 min 1 hr

What does this rate look like on a graph?

We can represent all kinds of motion graphically as

a comparison.

We will start with distance vs time graphs that

show the motion of the object during a certain

period of time.

The Moving Man

PHET simulations

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.

total distanceAverage speed

total time

dv

t

Speed

• The average speed of an object is defined as the total distance traveled divided by the total time elapsed

– Speed is a scalar quantity

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

s = d t

Speed equation

Average speed totally ignores

any variations in the object’s

actual motion during the trip.

The total distance and the total

time are all that is important

calculate

s = = d

t

20 m

4 s

s = 5 m/s

Scalar – magnitude only, not direction dependent!

A

B s = 20 m

Time t = 4 s

1) Average speed – average of all your speeds over the whole trip. For

example, on a trip to Florida, (910 miles) the traveler took 13 hours to get there.

The average speed would be 70 mph.

This doesn’t mean that the car was going exactly 70 mi/hr the entire time.

Sometimes the car was going faster, and others slower.

2) Constant Speed - traveling at the same rate for a long period of time. Constant speed is having the cruise control on in the car. The car maintains the

same speed the entire time you are clocking it.

3) Instantaneous speed - rate at which an object is traveling at a certain

moment. This is your speedometer in your car. It tells you how fast the car is

going at the time you look at it.

There are three types of speed:

velocity

A quantity that measures how fast an

object’s position has moved 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

D=12 m

20o

Velocity

• It takes time for an object to undergo a displacement

• The average velocity is the rate at which the displacement occurs

• generally use a time interval, so ti = 0

f iaverage

f i

x xxv

t t tV = ∆d = d1 – d0

∆t t1 – t0

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)

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.

- +

- +

+

Velocity can be +/-

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.

Recap:

Speed vs. Velocity

• Cars on both paths have the same average velocity since they had the same displacement in the same time interval

• The car on the blue path will have a greater average speed since the distance it traveled is larger

example problem

During a race, Carla covers 650 m in 125 s

running east on a straight road. Find Carla’s

average speed.

s = 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 360 s = 1.87 km

s hr hr

1.87 km = 5 km = 2.67 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.

W E

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:

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?

?check your understanding

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?

explain

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?

acceleration

Think about this....

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

Speed up

Slow down

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 speed or velocity in a given time.

• If velocity - must include a direction (vector).

• Any time an object’s speed or velocity is

changing, the object has an acceleration.

If a car is moving at a

constant velocity, then

its acceleration is zero!

acceleration = change in velocity = ∆v

change in time ∆t

acceleration = (vf) - (vi) time elapsed t

v

tt

vva

12

12

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

Acceleration is represented in (m/s2)

In 1 D motion, we can use vector arrows (or a +/-

sign) to show that something is speeding up or

slowing down.

V is positive and A is negative

V is positive and A is positive

in one-dimensional motion we only need the sign

to show when something is speeding up or slowing

down.

here is slowing down:

- the sign of the velocity and the acceleration is the

same if the object is speeding up and

- the sign of the velocity and the acceleration is the

opposite if the object is slowing down.

Review of Symbols and Units

• Displacement meters (m)

• Velocity/speed; meters per second (m/s)

• Acceleration; meters per s2 (m/s2)

• Time; seconds (s)

Make sure you convert any units when solving

problems so they are all the same!

try solving

Find the acceleration from the first 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

another problem to solve

A meteoroid changed velocity from 1.0

km/s to 1.8 km/s in 0.03 seconds. What is

its acceleration?

26.7 m/s2 (1.8) – (1) =

0.03 s

While racing out of our school parking lot, I time myself at

a speed of 40 meters per second seven seconds after

starting. What was my acceleration during this time?

Acceleration = vf – vi

t2 – t1

(40 m/s) / 7 s = 5.7 m/s2

If I were to accelerate at this rate for another

ninety seconds, how fast would I be going?

a = v/t so v = at (5.7 m/s2)(97 sec) = 550 m/s

Try some on

your own…

What would a velocity vs time graph look like?

Graphing velocity and

acceleration

Moving Man #2

Maze Game