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 B is not equal to the displacement from A to C. CA AB d d AC AB d d tance is an example of a scalar quantity. placement is an example of a vector quantity. ars have numerical value only (one number). ors have magnitude and direction (at least two numbers). A Example 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
4. Distance and displacement (displacement as an example of a vector)
Feb 09, 2016
4. Distance and displacement (displacement as an example of a vector). B. 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 B is not equal to the displacement from A to C. A. C. - PowerPoint PPT Presentation
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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
abd
dcba
Subtraction
cba
ca
b
b
ca
b
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 largestresultant 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
yr
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
ttdd
tdv
Definition:(total distance over total time)
b) Average velocity
Definition: (total displacement over total time)
initialfinal
initialfinal
ttrr
trv
6
initialfinal
initialfinalx tt
xxtxv
x-component of velocity:
7. Instantaneous speed and velocity(Speed and velocity at a given point)
txv
tx
0
limtrv
t
0lim
vv The magnitude of instantaneous velocity
is equal to the instantaneous speed
tdv
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
txvx
tantx
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
txv
tx
0
lim
8. Acceleration
vvtrv
trv
t
0lim
aatva
tva
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.
thmi
thmihmia
/5/5/10
thmi
thmihmia
/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
/22
2
0
0
vxstsma
smvmx
smvssmsmv
mxssmssmmx
/8 2/3/2
12 2
2/32/22
2
22
2
2
0
20
20
vvtxv
vvxxa
Example 2:
?/3
/210
2
0
vsma
smvmx 2
022
020
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
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