Chapter 2: 1-D Kinematicssrjcstaff.santarosa.edu/~lwillia2/20/20ch2_f14.pdfChapter 2: 1-D Kinematics...

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Chapter 2: 1-D Kinematics

Translational Motion Circular Motion

Projectile Motion Rotational Motion

Types of Motion

Natural Motion

•Objects have a proper place

•Objects seek their natural place

•External forces must be constantly

applied to moving objects in order

to keep them going.

•The heavier the object,

the faster it falls.

•Did not experiment to test theories.

Galileo Challenged The Dogma

Of Natural Motion with

Experiments

The natural motion of

a body is to remain in

whatever state of

motion it is in unless

acted upon by net

external forces.

Galileo Challenged The Dogma

Of Natural Motion

Galileo Challenged Aristotle Physics

In a vacuum, all objects fall with the same

acceleration due to gravity: 9.80 m/s2,

independent of their weight.

Galileo’s Motion Studies

gave us…

Definitions:

Distance and Displacement

(delta) means "change in"

= 'final - initial'

The total distance traveled relative to an origin.

Distance is a scalar.

Displacement is a vector. The unit is the meter.

0fx x x

An ant zig-zags back and forth on a picnic table as shown.

The ant’s distance traveled and displacement are

A. 50 cm and 50 cm.

B. 30 cm and 50 cm.

C. 50 cm and 30 cm.

D. 50 cm and –50 cm.

E. 60 cm and –30 cm.

QuickCheck 2.1

An ant zig-zags back and forth on a picnic table as shown.

The ant’s distance traveled and displacement are

A. 50 cm and 50 cm.

B. 30 cm and 50 cm.

C. 50 cm and 30 cm.

D. 50 cm and –50 cm.

E. 60 cm and –30 cm.

QuickCheck 2.1

Average Speed &Velocity

Speed is how fast something moves.

The average speed is the total distance per time.

The average velocity is the the total displacement per time.

Velocity is a vector. The unit is m/s.

total displacement

total time

Sense of Speed

1 / 3.6 / 2.24 /

10 / 36 / 22.4 /

20 / 72 / 44.8 /

30 / 108 / 67.2 /

m s km hr mi hr

1 / 2.25 /m s mi hr

1 / 0.62 /km hr mi hr

Acceleration How fast How fast is changing.

The rate at which the speed is changing.

Speeding up

Slowing down

Constant speed, changing direction.

change in velocity

change in time

Acceleration is in the direction of

the net Force but not necessarily

in the direction of velocity. Velocity is always in the direction of the motion!

Quicky Question

An automobile enters a freeway on-ramp at 15.0m/s and accelerates

uniformly up to 25.0 m/s in a time of 10.0s.

a) What is the automobile’s average velocity?

15 / 25 /20 /

m s m sm s

Which equation?

b) What is the automobile’s average acceleration?

f iv vva

225 / 15 /1 /

m s m sm s

Which equation?

Quicky Question

c) What is the distance traveled in this amount of time?

Which equation?

Quicky Question

200x m

(20 / )(10 )x

v x v t m s st

Motion Diagrams

Draw the Motion Diagram

Skiing through the woods

Here is a motion diagram of a car speeding up on a straight

The sign of the acceleration ax is

A. Positive.

B. Negative.

C. Zero.

QuickCheck 2.13

Here is a motion diagram of a car speeding up on a straight

The sign of the acceleration ax is

A. Positive.

B. Negative.

C. Zero.

QuickCheck 2.13

Speeding up means vx and ax have the same sign.

Which position-versus-time graph represents the motion shown in the

motion diagram?

Which position-versus-time graph represents the motion shown in the motion diagram?

fv vx vv v a

gave us…

Kinematic Equations

With a little al-jbr….

fv vx vv v a

Start:

0fv v a t

Assume constant acceleration!

0fv v a t

fv vx vv v a

Start:

0fv v a t

Assume constant acceleration!

0fv v a t

fv vx vv v a

Start:

0 0( )

f ix x v v a t

2f ix x v t a t

fv vx vv v a

t xv v

Start:

Combine &

Eliminate t:

t xv v a

0 2fv v a x Algebra:

v vx vv v a

v v at

x v t at

v v a x

gave us…

Kinematic Equations

Problem Solving Strategy

Acceleration: Changing Velocity From t = 0, how long does it take the car to come to a full stop?

How far does the car travel before it comes to a stop?

Draw the Motion Graph

Acceleration: Changing Velocity

Knowns

f iv vt

0 28 /5.6

f iv v at

Which equation to use?

Solve for t:

5.6t s

Acceleration: Changing Velocity

From t = 0, to t = 5.6s, how far does the car travel before it

comes to a stop?

Knowns

Which equation? 2

2x v t at

2 2128 5.6 ( 5 / )(5.6 ) 78.4

mx s m s s m

78.4x m

YOU TRY IT!

Brake Question You are driving a car going 80 km/hr (50mph) when a head-on

collision happens 25 meters ahead of you. If you can brake at 6 m/s2,

will you hit the crash or stop before it?

0 2fv v a x 2

(22 / )40.3 25

2( 6 / )

m sx m m

0: 80 / 22 / , 0, 6 / fKnowns v km hr m s v a m s

: ?Unknown x

CRASH!

Stopping

Distance

goes as the

SQUARE

of the

speed!

Stopping Distance

Traveling at 70 miles per hour, what is your breaking distance?

If v doubles,

d quadruples!!!

Stopping

Distance

goes as the

SQUARE

of the

speed!

0 2fv v a x

Motion Graphs

What kind of motion is this?

4001 /

mv m s

2 0 /v m s

4002 /

mv m s

What is the velocity during each

segment?

v vx vv v a

v v at

x v t at

v v a x

gave us…

Definitions of averages

Kinematic Equations with constant acceleration

Constant vs Changing Acceleration

Depends on the FORCE

Constant Forces

• Constant pushes and pulls

• Inclined planes

• Gravity near the earth (Free Fall)

• Pulleys, Conical Pendulums

Variable Forces

• Springs and Pulleys

• Air Resistance

• Gravity Far from Earth

• Electricity and Magnetism

• MOST FORCES!!!!

Free Fall Unless told otherwise, ignore air resistance for

free fall problems!

Galileo Challenged Aristotle Physics

In a vacuum, all objects fall with the same

acceleration due to gravity: 9.80 m/s2,

independent of their weight.

Acceleration of Freely Falling Object

• The acceleration of an object in

free fall is directed downward,

regardless of the initial motion

• The magnitude of free fall

acceleration is g = 9.80 m/s2

g decreases with increasing altitude

– g varies with latitude

– 9.80 m/s2 is the average at the Earth’s

surface

– We will neglect air resistance

– g is a SCALAR!!! POSTIVE

Free Fall Equations For any object in the absence of air resistance.

29.80 /ya g m s

Customize:

v v gt

y v t gt

v v g y

Kinematic Eqs:

v v at

x v t at

v v a x

Note: v0 can be negative!

(taking up as +y)

Falling from Rest

2y at t

~ 10 /

Estimate

a g m s

10v at t

fv v gt

y v t gt

!v y How FAR is not

How FAST!

as +y:

How Far: y(t) ~ t2

0fv v at

2y v t at

How Fast: v(t) ~ t1

How Fast How Fast is

Changing! 29.80 /g m s

FIRST: Define Reference Frame In this reference frame,what is the sign of a? 29.80 /a m s

What is v at t = 3s?

0fv v at

20 9.80 (3 )

0: 0, 9.8 / , 3Knowns v a m s t s

: ?fUnknown v

Negative because it is moving downward, in the negative direction!

FIRST: Define Reference Frame

0: 0, 9.8 / , 3 , 29.4 /fKnowns v a m s t s v m s

: ?Unknown y

The displacement is negative because it is moves downward, in the negative

direction but “how far” is a distance – a scalar – and is positive!

How far did the ball fall in those 3 seconds?

2y v t at

20 ( 9.8 )(3 )

The ball fell 44.1m.

Throwing up is Also Free Fall!

Symmetry of G Field.

~ 10 /

Estimate

a g m s

fv v gt

y v t gt

What Goes Up Must Come Down

Someone standing at the

edge of a cliff throws one

ball straight up and one

straight down at the same

speed. Ignoring air

resistance, which ball

strikes the ground with the

greatest speed?

Free Fall Question: You throw the rock down with an initial speed of

30 m/s. The rock hits the ground in 3 seconds. With what

speed will the rock hit the ground?

230 9.8 (3 )

0fv v at

How high is the cliff?

0: 30 / , 9.8 / , 3Knowns v m s a m s t s

: ?fUnknown v

Free Fall

2y v t at

2( 30 / )(3 ) ( 9.8 / )(3 )m s s m s s

The cliff is 134 m high.

0: 30 / , 9.8 / , 3Knowns v m s a m s t s

: ?Unknown y +y

Question: You throw the rock down with an initial speed of

30 m/s. The rock hits the ground in 3 seconds. With what

speed will the rock hit the ground? How high is the cliff?

Free Fall: Throwing Up What is the speed at the top of the path?

What is the acceleration at the top?

a = -9.80 m/s2

What is the velocity at the same height

on the way down?

-30 m/s

With what velocity will

the rock hit the ground?

-59.4 m/s

SAME as if you threw it

straight down at 30m/s!

How long does it take to hit the ground? First try to guess!

0fv v at

59.4 / 30 /

fv v m s m st

9.12t s

0: 30 / , 9.8 / , 3 , 59.4 /fKnowns v m s a m s t s v m s

: ?Unknown t

How long to the top? How long back to launch point? Final v increases by 30m/s?

I guess about 9 seconds!

Free Fall: Throwing Up Problem

A ball is tossed straight up in the air. At its very

highest point, the ball’s instantaneous acceleration ay

A. Positive.

B. Negative.

C. Zero.

QuickCheck 2.18

A ball is tossed straight up in the air. At its very

highest point, the ball’s instantaneous acceleration ay

A. Positive.

B. Negative.

C. Zero.

QuickCheck 2.18

Figure (a) shows the motion diagram of an object sliding down a straight, frictionless inclined plane.

Figure (b) shows the the free-fall acceleration the object would have if the incline suddenly vanished.

This vector can be broken into two pieces: and .

The surface somehow “blocks” , so the one-dimensional acceleration along the incline is

The correct sign depends on the direction the ramp is tilted.

Motion on an Inclined Plane

The ball rolls up the ramp, then back down. Which is the correct acceleration graph?

Here is a motion diagram of a car moving along a straight road:

Which velocity-versus-time graph matches this motion diagram?

QuickCheck 2.5

Here is a motion diagram of a car moving along a straight road:

Which velocity-versus-time graph matches this motion diagram?

QuickCheck 2.5

Rank in order, from largest to smallest, the accelerations a

A– a

points A – C.

A) aA > a

> aC > a

> aA > a

E) aC > aB > aA

A) aA > a

> aC > a

> aA > a

E) aC > aB > aA

Rank in order, from largest to smallest, the accelerations a

A– a

points A – C.

Here is a position graph

of an object:

At t = 3.0 s, the object’s

velocity is

A. 40 m/s.

B. 20 m/s.

C. 10 m/s.

D. –10 m/s.

E. None of the above.

QuickCheck 2.7

Here is a position graph

of an object:

At t = 3.0 s, the object’s

velocity is

A. 40 m/s.

B. 20 m/s.

C. 10 m/s.

D. –10 m/s.

E. None of the above.

QuickCheck 2.7

Which velocity-versus-time graph or graphs goes with this acceleration-versus-time graph? The particle is initially moving to the right and eventually to the left.

Which velocity-versus-time graph goes with the position-versus-time graph on the left?

Which position-versus-time graph goes with the velocity-versus-time graph at the top? The particle’s position at ti = 0 s is xi = –10 m.

Speedy Sally

Speedy Sally, driving at 30.0 m/s, enters a one-lane tunnel. She then observes a slow-moving van 155 m ahead traveling at 5.00 m/s. Sue applies her brakes but can accelerate only at 2.00 m/s2 because the road is wet. Will there be a collision? If yes, determine how far into the tunnel and at what time the collision occurs. If no, determine the distance of closest approach between Sally's car and the van. Sketch the x-t graphs for both the vehicles. What does it mean?

Rock Drop

A rock is dropped from rest into a well. The sound of the splash is heard 3.20 s after the rock is released from rest. How far below the top of the well is the surface of the water? The speed of sound in air (at the ambient temperature) is 336 m/s.

Chapter 2: 1-D Kinematicssrjcstaff.santarosa.edu/~lwillia2/20/20ch2_f14.pdfChapter 2: 1-D Kinematics...

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