Kinematics in One Dimension
Mechanics: The Study of Motion


Kinematics: How objects move
Dynamics: Forces and why objects move
Speed is Measured From a
Frame of Reference
Frames of Reference



A car moving at 60 mph looks as if it is
standing still if you are moving at 60 mph.
How fast does it seem to move if you are
going 30 mph in the same direction as the
car?
How fast does it seem to move if you are
moving 60 mph in the opposite direction?
Frames of Reference



Any measurement of position, distance or
speed must be made with respect to a
frame of reference
The motion of an object is highly
dependent on where you observe it from
Inside a pane flying at constant velocity, if
there were no windows could you tell you
were moving? How?
Measuring Motion



The displacement of an object is defined
as the change of position of an object
Displacement is different from the distance
an object travels? How?
Displacement is a vector quantity


It has magnitude and direction
Displacement over a unit of time is velocity
Displacement example
30 m west
70 m east
Net displacement = 40 m east
∆x = x1 – x2
Graphical Interpretation
Slope of the line is zero
Velocity is zero
Slope of the line is velocity
Velocity is negative
Distance
Time
Slope of the line is velocity
Velocity is positive
Graphical Interpretation
Slope of the line is zero
Acceleration is zero
Slope of the line is velocity
Acceleration is negative
Velocity
Time
Slope of the line is
acceleration
Acceleration is positive
Average Speed & Velocity

Average speed = distance traveled
elapsed time

Average velocity = displacement
elapsed time
v=
x2 –x1
∆x
t2 - t1 = ∆t
Constant Velocity (D vs T)
What happens when the lines cross?
Constant Velocity (V vs T)
Why don’t the lines cross?
Instantaneous Velocity
If
v =
∆x
∆t
Then, the instantaneous velocity is:
∆x
v = lim
∆ 0
∆t
Instantaneous Velocity
As we let ∆t get smaller and smaller
the line whose slope we use to get
the velocity looks more and more like
a tangent to the curve. In the limit of
∆ → 0 the line becomes the
derivative of the curve.
∆x
∆t
Acceleration


Acceleration is the change in velocity of an
object
Any change in velocity is the result of an
acceleration
Avg Accel = Final velocity – Original velocity
Time
Calculating Acceleration

A car accelerates from a stop. After 6
seconds it is traveling at 28 m/s (about 60
mi/hr). What was its average
acceleration?



Change in speed = 28 m/s
Time = 6 seconds
Acceleration = (28 m/s) / 6 s
= 4.67 m/s/s = 4.67 m/s2
Motion to the Right with Constant
Rightward Acceleration
Motion to the Right with Constant
Leftward Acceleration
Equations of Constant Acceleration
v = v0 + at
x = x0 + v0t + ½ at2
v2 = v02 + 2a(x – x0)
v = v + v0
2
Falling Objects


The most common example of constant
acceleration is an object falling towards the
earth
The acceleration due to gravity is 9.8 m/s2


At the end of each second of fall the speed of the
object will increase by 9.8 m/s
NOTE: on the AP test multiple choice problems
assume that the gravitational acceleration
is 10 m/s2
Questions

A ball is thrown upward



What is the magnitude and
direction of its acceleration
at A?
What is the magnitude and
direction of its acceleration
at B?
What is the direction of its
velocity at A and B?
B
A
Multiple Choice

P. 44
Classwork/Homework

pp. 43, #21, 25, 27, 41, 47, 53, 57
Classwork




Go to http://cwx.prenhall.com/giancoli/
Select Chapter 2, then push Begin
Select Practice Questions
Answer the 25 questions and then push Submit
for Grading at that time you can enter your name
and my email address:


jtimson@bpi.edu
It will save you time in the future if you set up an
account in your name
Which Skier Gets There First?
http://www.upscale.utoronto.ca/GeneralInter
est/Harrison/Flash/ClassMechanics/RacingS
kiers/RacingSkiers.html
Derivation of Equations of Linear
Motion
Effects of Constant Acceleration
Practice:


Complete the multiple choice questions
from yesterday.
Work with your group to brainstorm
answers to the concept questions on pp.
45-46. Be prepared to discuss your
thoughts!
Practice!

Work on your hw!!!
Do Now (9/3/13): (on a new sheet)
An object is launched with initial velocity 20
m/s at an angle of 30°. Find the :
1. Initial vertical velocity
2. Initial horizontal velocity
3. Maximum height
4. Time of flight
5. How far away the object landed
Two-dimensional Motion
30 m/s



An object is dropped off a 40 meters
cliff. How long does it take to reach
the ground?
The same object is thrown horizontally
with a velocity of 30 m/s. How long
will it take to fall to ground?
Velocity is a vector. The horizontal
velocity has no bearing on the time it
takes to fall to the ground. All it does
is change the trajectory
40 m
Vector Problems
How long does it
take to cross the
river?
2 km/hr
5 km/hr
10 km
If the river is
flowing at 2 km/hr,
how long does it
take to cross the
river?
Vector problems (cont’d)
If the river is
flowing at 2 km/hr,
how far
downstream will
the boat be?
2 km/hr
? km
5 km/hr
10 km
If the crew wanted to
end up directly across
the river, what path
should they follow?
How long will it take
them to cross the river
now?
Practice:


Work with your group to brainstorm answers
to the conceptual questions 1-8 on p. 71
10 min!
Do Now (9/4/13):


In 1974 Nolan Ryan pitched a baseball at
100.8 mph. If a pitch were thrown
horizontally with this velocity, how far
would ball fall vertically by the time it
reached home plate 60 ft away?
(*hint – conversions are in your textbook!!)
Practice:



Work with your group to brainstorm
answers to the conceptual questions 9
and up on p. 71
10 min!
14, 16, 20
Practice:


Complete problems 21 and 27 in Chapter 3.
Problem 60 is a bonus!
Do Now (9/5/13):


One baseball is dropped from a height,
while another is launched horizontally from
the same height. Draw a diagram to show
their motion throughout their respective
trips.
How far apart (timewise) will they land?
Agenda



Homework
Quiz info
Review: work on your homework,
classwork (21, 27, and *60), conceptual
questions, and/or your notecard
Do Now (9/6/13):



Come in quietly, pass in your Do Now’s,
then clear your desk of everything except
your quiz materials
No sharing notecards
No sharing calculators
Forces Are Vectors Also
200N
120N
53o
150N
x direction
-120 N + 0.6 (200 N) = 0
y direction
-150 N + 0.8 (200 N) = 10 N
cos 53o = 0.6
sin 53o = 0.8
Momentum
Momentum

Momentum depends on the mass of an
object and the speed it is going.


Momentum = mass x velocity
Because velocity has direction then
momentum does, also.
Momentum of Objects

Put the following in the order of most
momentum to least:





Mosquito
Automobile
Space Shuttle
Bullet
Freight Train
Questions


Does a small object always have less
momentum than a large one?
How can a rifle bullet knock over a person
or an animal?
Conservation of Momentum


When two objects collide, the momentum
after the collision must be equal to the
momentum after the collision.
The total momentum of any group of
objects remains the same unless outside
forces act on the objects.
Conservation of Momentum
The momentum of the two astronauts is equal to the
momentum of the first astronaut before the collision
Conservation of Momentum—
Inelastic Collisions
Before the collision momentum = 1000 kg x 20 m/s = 20,000 kg m/s
After the collision momentum = (1000 kg + 3000 kg) x 5 m/s
= 20,000 kg m/s
Conservation of Momentum—
Elastic Collisions
After the collision the total momentum of the two vehicles is
the same as the car’s before the collision.
Conservation of Momentum—
Elastic Collisions
Conservation of Momentum—
Inelastic Collisions
Conservation of Momentum—
Elastic Collisions
Forces
Force

What causes objects to move?



FORCE
A force is a push or a pull
A force can make an object stop or start
moving or change its speed or direction.
Newton’s Laws
Newton’s First Law of Motion

An object at rest will remain at rest and an
object in motion will remain in motion at a
constant velocity unless acted upon by an
unbalanced force.
Inertia



The tendency of objects to remain at rest
or remain in motion is called inertia.
The effects of inertia cause you to move
forward when a car stops quickly.
If you are standing in a bus or a subway
car when it starts up you move backward
because your body’s inertia wants to
“remain at rest.”
Newton’s First Law
The passenger remains in motion when the car stops unless
he is acted upon by a force such as a seatbelt or a wall.
Crash Test Dummies
Newton’s First Law
When the motorcycle stops, the rider continues his motion.
Real Life Demo




Grabber 4/23
Page 328: Define friction
Reading through Section 13-2, give 3 examples
of friction.
Would we be able to walk if there was no friction?
If you managed to get started, could you stop?
Sand is sometimes put on icy roads and
sidewalks. Why does this help walking and
driving?
Newton’s First Law

An object at rest will remain at rest and an
object in motion will remain in motion at a
constant velocity unless acted upon by an
unbalanced force.
Newton’s First Law
Warm-up 9/15
1.
2.
3.
Force = mass x acceleration
A 5.5 kg watermelon is pushed across the
table. If the acceleration is 4.2 m/sec/sec to
the right, find the net force on the melon.
Astronauts in the space shuttle experience an
acceleration of about 35 m/sec/sec during
liftoff. What is the force on a 75 kg?
A 6.0 kg object undergoes an acceleration of
2.0 m/sec/sec. What is the net force acting on
it? If this same force is applied to a 4.0 kg
object, what is the acceleration produced?
Newton’s Second Law
Newton’s Second Law


The second law of motion show how force,
mass, and acceleration are related.
Force = mass x acceleration
When mass is measured in kilograms and
acceleration is in meters/second/second, the
force is measured in newtons. (N).

One newton is the force required to accelerate one
kilogram of mass at one meter/second/second.
1 N = 1kg x 1m/sec/sec
Second Law Diagram
Example




How much force is needed to accelerate a
1400 kg car 2 meters/second/second?
Force = mass x acceleration
Force = 1400 kg x 2 meters/second/second
Force = 2800 kilogram-meters/second/second
= 2800 N
Problems

How much force is needed to accelerate a 66 kg
skier by 1 m/sec/sec?


Force = mass x acceleration
= 66 kg x 1 m/sec/sec
= 66 kg m/sec/sec = 66 N
What is the force on a 1000 kg elevator that is
falling freely at 9.8 m/sec/sec?

Force = mass x acceleration
= 1000 kg x 9.8 m/sec/sec
= 9800 kg m/sec/sec = 9800 N
Free Body Diagrams




1. Reduce the object to a single point.
2. Draw all forces as vectors with the tails
originating at the point (object).
Be sure to make the length of the vector
reflect the relative magnitude of the force.
The force vectors should be pointing in the
direction that the forces act
3. LABEL ALL FORCES
Forces acting on objects in one
dimension

A book is at rest on a
table top. Diagram
the forces acting on
the book.
When Forces Balance Velocity is
Constant
Forces acting on objects in one
dimensions

A flying squirrel is
gliding (no wing flaps)
from a tree to the
ground at constant
velocity. Consider air
resistance. Diagram
the forces acting on
the squirrel.
Friction




What force causes a car to slow to a stop
if the engine is turned off?
What force keeps a car on a NASCAR
track in the corners?
Friction is a force that opposes motion
The amount of friction depends on the
type of surfaces in contact.
Frictional Force Depends on the
Weight of an Object and the Surface
• The frictional
force on an object
depends on the
“normal force”
and the nature of
the surface.
• Friction always
opposes the
direction of
motion
The frictional force on a object
of mass m is given by:
Ffrict = μ x mg
Where μ is the “coefficient of friction” for
the surface (0 < μ < 1
Friction questions

Which has more friction?



Sliding a 2 kg brick across a table or sliding
two bricks stacked on each other?
An icy road at 10o or one at 32o?
If a block of rubber is slid across a table
quickly, its surface will get warm. Where
does the heat come from?
Forces in two dimensions

A rightward force is
applied to a book in order
to move it across a desk
with a rightward
acceleration. Consider
frictional forces. Neglect
air resistance. Diagram
the forces acting on the
book
Newton’s Third Law
Question

Imagine you are an astronaut on a spacewalk outside the space shuttle.



You have used up all the gas in your jet pack.
How do you get back to the shuttle?
ANSWER: Throw the jet pack away from
the shuttle and you will go towards it.
Newton’s Third Law

Whenever one object exerts a force on
another, the other exerts and equal force
back in the opposite direction.

For every action there is an equal but
opposite reaction.
Explain These Examples




Rowing a boat
Birds flying
Rockets
A book sitting on a table
Review



What is inertia? How is it involved in
Newton’s first law of motion?
What three quantities are related in
Newton’s second law of motion? What is
the relationship among them?
What does Newton’s third law say about
action-reaction forces?
Gravity
Galileo, Newton, and Gravity

Galileo was born in 1564.




People of his time believed that heavy objects would fall
faster than light ones.
Galileo proved that objects fall at the same rate (assuming
air resistance is not significant).
A falling object is accelerating (it gets faster as it
falls)
According to Newton’s second law, an accelerating
object must have a force acting on it
Therefore….


The acceleration of a falling object is due
to the force of gravity between the object
and the earth.
At the earth’s surface every object
accelerates at a rate of 9.8 m/sec/sec.
 This is the gravitational acceleration,
which is abbreviated “g”
Gravitational Acceleration

When an object is dropped from a
mountain or tall building



At the end of the first second its velocity is 9.8
m/sec
At the end of two seconds its velocity is 19.6
m/sec
At the end of three seconds it will have a
velocity of 29.4 m/sec.
Air Resistance





Do a leaf and a piece of paper fall as fast
as a rock?
The reason they don’t is air resistance.
All falling objects encounter air resistance.
Sometimes the air resistance is enough to
keep an object from accelerating faster.
When this happens we call the speed it
achieves “terminal velocity.”
Universal Gravitation


Newton realized that the forces acting on
falling objects on earth were no different
from those forces that keep the moon
orbiting the earth or the earth orbiting the
sun.
His law of universal gravitation states that
all objects in the universe attract each
other with the force of gravity.
Effects of Gravity

When objects have more mass they have more
gravitational force.


Large planets have stronger gravity than small
planets.
The force of gravity is relatively small and is
dependent on the mass of the two objects
attracting each other.

Your textbook doesn’t jump into your hand because
you and the book don’t generate enough gravitational
force
Weight and Mass
Weight is a measure of the force of gravity on an
object.
weight = mass x acceleration due to gravity
=mxg





Weight varies with the distance from the center of the
earth.
Weight varies with what planet you are on
The mass of an object does not change regardless of
where it is measured.
The “official” unit of weight is the newton
Weight vs. Mass


Mass is often confused with weight.
Mass is a measure of how much matter is
in an object.


The more matter, the more mass!
The pull of gravity on an object determines
its weight.
CRES Review

The space shuttle has a mass of 2 million
kg. At lift off its engines produce an
upward force of 30 million newtons.



What is the weight of the shuttle? (w = m x g)
What is the acceleration at launch?
The average acceleration during the ten
minutes of engine burn is 13 m/sec/sec.
What speed does it achieve?