4. Distance and displacement (displacement as an example of a vector)

A

B

C

Distance - fundamental physical quantity measured in units of length.

Displacement - physical quantity that should be described by both its magnitude (measured in units of length) and direction.

Example 1: The distance between points A and B is equal to the distance between A and C.

In contrast, the displacement from point A to point Bis not equal to the displacement from A to C.

CAAB dd

ACAB dd

Distance is an example of a scalar quantity.Displacement is an example of a vector quantity.

Scalars have numerical value only (one number).Vectors have magnitude and direction (at least two numbers).

AExample 2: For the motion around a closed loop (from A to A) the displacement is zero, but the distance is not equal to zero.

1

•A vector has magnitude as well as direction•Some vector quantities: displacement, velocity, force, momentum•A scalar has only magnitude and sign•Some scalar quantities: mass, time, temperature

5. Vectors

Geometric presentation: a

Notations: - letter with arrow; a – bold fonta

aa

Magnitude (length of the vector):

A

Some properties: B

CBA

C

2

5a. Vector addition (geometric)

c

ab

cba

Two vectors:

Several vectors

c

ab

c

a b

d

dcba

Subtraction

cba

cab

b

cab

b

3

Question 2: A person walks 3.0 mi north and then 4.0 mi west. The length and direction of the net displacement of the person are:

1) 25 mi and 45˚ north of east2) 5 mi and 37˚ north of west3) 5 mi and 37˚ west of north4) 7 mi and 77˚ south of west

Question 3: Consider the following three vectors:What is the correct relationship between the three vectors?

BA

Question 1: Which of the following arrangements will produce the largest

resultant when the two vectors of the same magnitude are added?

B C

BAC

BAC

.2

.1 BAC

BAC

.4

.3

A

β

β = 37˚<45˚ϴ= 53˚> 45˚

4

5b. Vectors and system of coordinates

x

y

r

xrx

yry

zyxrrrrrrr zyxzyx ,,,,

yxrrrrr yxyx ,,

x

y

z

r

2D:

3D:

xr

yr r

5

6. Average speed and velocity

a) Average speed

initialfinal

initialfinal

tt

dd

t

dv

Definition:(total distance over total time)

b) Average velocity

Definition: (total displacement over total time)

initialfinal

initialfinal

tt

rr

t

rv

6

initialfinal

initialfinalx tt

xx

t

xv

x-component of velocity:

7. Instantaneous speed and velocity

(Speed and velocity at a given point)

t

xv

tx

0lim

t

rv

t

0lim

vv The magnitude of instantaneous velocity

is equal to the instantaneous speed

t

dv

t

0

limDefinition:

In contrast, the magnitude of average velocity is not necessarily equal to the average speed

7

a) Instantaneous speed

b) Instantaneous velocity

Definition:

6. Geometric interpretation

t1t 2t 3t 4t

1t

2t

1x

2x

t

xv

tant

x

vtxx 0

Velocity is equal to the slope of the graph (rise over run): distance over time.

x

8

Question: The graph of position versus time for a car is given above. The velocity of the car is positive or negative?

a) One dimensional uniform motion (v = const)

9

A

B

t

x

x

t

b) Motion with changing velocity

Question: The graph of position versus time for a car is given above. The velocity of the car is positive or negative? Is it increasing or decreasing?

Instantaneous velocity is equal to the slope of the line tangent to the graph.(When Δt becomes smaller and smaller, point B becomes closer and closerto the point A, and, eventually, line AB coincides with tangent line AC.)

C

t

xv

tx

0lim

8. Acceleration

vvt

rv

t

rv

t

0lim

aat

va

t

va

t

0lim

•Acceleration shows how fast velocity changes•Acceleration is the rate at which velocity is changing - “velocity of velocity”

10

Example: The speed of a bicycle increases from 5 mi/h to 10 mi/h.In the same time the speed of a car increases from 50 mi/h to 55 mi/h.Compare their accelerations.

Hence, the acceleration of the bicycle is equal to the acceleration of the car.

t

hmi

t

hmihmia

/5/5/10

t

hmi

t

hmihmia

/5/50/55

Solution:

We denote the time interval as Δt. Then the acceleration of the bicycle is:

and the acceleration of the car is:

11

9. Motion with constant acceleration

atvv

attvxx

0

2

00 2

Example 1:

? ?

2

/3

/2

2

2

0

0

vx

st

sma

smv

mx

smvssmsmv

mxssm

ssmmx

/8 2/3/2

12 2

2/32/22

2

22

2

2

0

20

20

vv

t

xv

vvxxa

Example 2:

?

/3

/2

10

2

0

v

sma

smv

mx 20

220

20

20 2 22 vxavvvxavvxxa

smvsmmsmv /8 /210/32222

12

13

Question 2: If the velocity of a car is zero, can the acceleration of the car be non-zero?

A) Yes B) No C) It depends

Question 1: If the velocity of a car is non-zero, can the acceleration of the car be zero?

A) Yes B) No C) It depends

Question 3: The graph of position versus time for a car is given below. What can you say about the velocity of the car over time?

A) It speeds up all the timeB) It slows down all the timeC) It moves at constant velocityD) Sometimes it speeds up and sometimes it slows downE) Not really sure

x

t