1. Measurements & units2. Scalars & vectors3. Displacement, Velocity and

acceleration4. Relative velocity.5. Motion in two dimensions and

in three dimensions6. Special case: Gravity

Summary• Vectors: Positions, Displacement,

Velocity and Acceleration.

Vector or Scalar?Vector or Scalar?

• Speed………..

• Velocity……...

• Acceleration..

• Time………….

• Force…………

• Distance……..

scalar

vector

vector

scalar

scalar

it depends...

Some Derivatives

• Powers

• Trig Functions

• Exponentials

?xn dx

d

?)xsin( dx

d

?ex dx

d

Average Velocity What is the average velocity in the last

second (t = 3 to 4) ?

A. 2 m/sB. 4 m/sC. 1 m/sD. 0 m/s

x (meters)

t (seconds)

2

6

-2

4

1 2 43

Instantaneous velocity What is the instantaneous velocity in the

last second?

A.-2 m/sB. 4 m/sC. 1 m/sD. 0 m/s

x (meters)

t (seconds)

2

6

-2

4

1 2 43

Average Speed What is the average speed over the

first 4 seconds ?

A. 2 m/sB. 4 m/sC. 1,5 m/sD. 0 m/s

x (meters)

t (seconds)

2

6

-2

4

1 2 43

turning point

Correcting home exercises

What is displacement of a train from staring point to point at 3 seconds after ?

What is the velocity and acceleration of a train ?? (AV or IV) from staring point to point at 3 seconds after ?

Part 4

Relative velocity.

Air speed

Ground speed

Brainstorming

Galilean formula of velocity sum

Learning Check

Solution

a) Up

Learning Check

ground

river

boat

Part 5

Motion in one dimension and in two dimensions

Linear motion

0 is certain point

Green car with solar cell

Learning Check

John is moving to x direction by equation: X= - 25t2 +3t +7 (cm)

1- What is John ‘s position at time t=0? and t = 3(s) ? 2- What is his velocity at time time t=0? and t = 3(s) ? Average speed of John after 10s

moving? 3- What is his acceleration at time time t=0? and t = 3(s) ? Average acceleration of John after 10s

moving?

Learning checkAcceleration vs Time Plots

• Gives acceleration at any time.

• Area gives change in velocity Acceleration at t=4, a(4) =

Change of v between t=4 and t=1. v =

a

t4

3

-3

Constant Acceleration

Equation of motion isdt

dva

dvdt a

dtadv

tavv o

The o in v subscript refers to the original or initial value at the beginning of the time interval of interest.

Integrate both sides

where acceleration is constant.

Solution

Arranging this equationdt

dxv

dxdtv

dttavdx oo

dttavdx oo

2ooo ta

2

1tvxx

Substituting the velocity equation from the previous page

Integrating both sides

Learning Check

X=2+10t +4t2 (m)At t=3 x= 2+3.10 +4.9 = 68 (m)V= 10 + 8t (m/s)At x=4 4= 2+10t +4t2 4t2 +10t –2 =0 t = ?

Part 5

Motion in two dimensions

Motion in two dimensions

2y

2x

2 VVV 2y

2x

2 aaa

Positions in 2 dimensions

-100

-80

-60

-40

-20

0

20

0 5 10 15 20

v (m/s)

t (seconds)

-300

-200

-100

0

100

0 5 10 15 20

x (meters)

t (seconds)

• Where is velocity zero?• Where is velocity positive?• Where is velocity negative?• Where is speed largest?

• Where is acceleration zero?• Where is acceleration positive?

position vs. time

velocity vs. time

Example

Learning Check

Learning Check

Part 6

Free fall

Isaac Newton in 1689, by Sir Godfrey Kneller.

History

Learning Check

Exercises of today’s lecture

A ball is thrown straight up in the air and returns to its initial position. During the time the ball is in the air, which of the following statements is true?A - Both average acceleration and average velocity are zero.B - Average acceleration is zero but average velocity is not zero.C - Average velocity is zero but average acceleration is not zero.D - Neither average acceleration nor average velocity are zero.

Summary of Concepts

• kinematics: A description of motion• position: your coordinates• displacement: x = change of position• velocity: rate of change of position

– average : x/t– instantaneous: slope of x vs. t

• acceleration: rate of change of velocity– average: v/t– instantaneous: slope of v vs. t

Class Question

• How do units differ from variables?

List 10 clear examples of units and 10 clear examples of variables.

Problem

• A motorcycle moves with an initial velocity of 30m/s.

• When its brakes are applied, it decelerates at 5.0m/s2 until it stops.

• Plot the position, velocity and acceleration as a function of time.

• What is the position, velocity and acceleration 2 seconds after the brakes are applied?

2ooo ta

2

1tvxx

tavv o

Use bike computer

Bikebrain

Source: http://www.bikebrain.com

Attaches to a “PalmPilot”

Problem

• A car starts from rest and travels northward.• It accelerates at a constant rate for 30

seconds until it reaches a velocity of 55mph.• Plot the acceleration, velocity and position

as a function of time.

Problem

• A girl shoots an arrow upward.

• It strikes the ground 10.0 seconds later.

• What was its initial velocity and what was the maximum height?

2yoyoo ta

2

1tvyy

tavv yyoy

Problem

• A man standing on a 20-m helicopter throws a ball upward at 120 m/s.

• How long does it take to hit the ground?

Team Exercise, 3 min.

1. The derivative of velocity with respect to time is:– position or acceleration

2. By integrating velocity with respect to time we get:– distance traveled or acceleration

3. The derivative of position with respect to time is:– acceleration or velocity

4. Integrating acceleration twice with respect to time is :– velocity squared or distance

5. The derivative is associated with the _________ while the integral is associated with _________– area under the curve, slope

Team Exercise (3 minutes)

• One dimensional motion– What is the distance traveled in 3 seconds? – What is the acceleration at 1.25 hours?

Sp

eed

, mp

h

3210

010

20

Time, hours

Please

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