kinematics in one dimension

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

70 m east30 m west

Net displacement = 40 m east

∆x = x1 – x2

Graphical Interpretation

Time

Distance

Slope of the line is velocityVelocity is positive

Slope of the line is velocityVelocity is negative

Slope of the line is zeroVelocity is zero

Graphical Interpretation

Time

Velocity

Slope of the line is accelerationAcceleration is positive

Slope of the line is velocityAcceleration is negative

Slope of the line is zeroAcceleration is zero

Average Speed & Velocity

Average speed = distance traveled elapsed time

Average velocity = displacement elapsed time

x2 –x1 ∆x t2 - t1 ∆tv = =

Constant Velocity (D vs T)

What happens when the lines cross?

Constant Velocity (V vs T)

Why don’t the lines cross?

Then, the instantaneous velocity is:

∆x ∆t

Instantaneous Velocity

∆x ∆t

v = lim

If

∆ 0

v =

Instantaneous Velocity

∆t

∆x

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.

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?

A

B

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/GeneralInterest/Harrison/Flash/ClassMechanics/RacingSkiers/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

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

30 m/s

Vector Problems

10 km

5 km/hr

How long does it take to cross the river?

If the river is flowing at 2 km/hr, how long does it take to cross the river?

2 km/hr

Vector problems (cont’d)

10 km

5 km/hr

If the river is flowing at 2 km/hr, how far downstream will the boat be?

2 km/hr

? 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

53o120N

150N

cos 53o = 0.6sin 53o = 0.8

x direction-120 N + 0.6 (200 N) = 0y direction-150 N + 0.8 (200 N) = 10 N

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/sAfter 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. 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.

2. Astronauts in the space shuttle experience an acceleration of about 35 m/sec/sec during liftoff. What is the force on a 75 kg?

3. 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?

Force = mass x acceleration

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

Ffrict = μ x mgWhere μ is the “coefficient of friction” for

the surface (0 < μ < 1

The frictional force on a object of mass m is given by:

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 space-walk 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 = m x g

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?