Quantities Units Vectors Displacement Velocity Acceleration Kinematics Graphing Motion in 1D Some Physics Quantities Vector quantity with both magnitude size and direction Scalar quantity with magnitude only ID: 643445
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Slide1
Physics Intro & Kinematics
QuantitiesUnitsVectorsDisplacement
Velocity
Acceleration
Kinematics
Graphing Motion in 1-DSlide2
Some Physics Quantities
Vector - quantity with both magnitude (size) and direction Scalar - quantity with magnitude only
Vectors
:
Displacement
Velocity Acceleration Momentum Force
Scalars:
Distance
Speed
Time
Mass
EnergySlide3
Mass vs. Weight
On the moon, your mass would be the same, but the magnitude of your weight would be less.Mass
Scalar (no direction)
Measures the amount of matter in an object
Weight
Vector (points toward center of Earth)
Force of gravity on an objectSlide4
Vectors
The length of the arrow represents the magnitude (how far, how fast, how strong, etc, depending on the type of vector).The arrow points in the directions of the force, motion, displacement, etc. It is often specified by an angle.
Vectors are represented with arrows
42°
5 m/sSlide5
Vectors vs. Scalars
One of the numbers below does not fit in the group: 35 ft 161 mph 70° F
200-m, 30° East of North
12 200 peopleSlide6
Vectors vs. Scalars
The answer is: 200-m, 30° East of NorthWhy is it different? Numbers w/ magnitude only are called SCALARS.Numbers w/ magnitude and direction are called VECTORS.Slide7
Vector Example
Particle travels from A to B along the path shown by the dotted red lineThis is the distance traveled and is a scalarThe displacement
is the solid line from A to B
displacement
is independent of path taken between two points
displacement is a vectorNotice the arrow indicating directionSlide8
Other Examples of Vectors
Displacement (of 3.5 km at 20o North of East)Velocity (of 50 km/h due North)Acceleration (of 9.81 m/s2 downward)Force (of 10 Newtons in the +x direction
)Slide9
Notation
Vectors are written as arrows. length describes magnitudedirection indicates the direction of vector…Vectors are written in bold text in your book
Conventions for written notation shown below…Slide10
Adding Vectors
Case 1: Collinear VectorsSlide11
What is the
ground speed of an airplane flying with an air speed of 100 mph into a headwind of 100 mph?Slide12
Adding Collinear Vectors
When vectors are parallel: just add magnitudes and keep the direction.Ex: 50 mph east +
40 mph
east =
90 mph eastSlide13
Adding Collinear Vectors
When vectors are antiparallel: just subtract smaller magnitude from larger - use the direction of the larger.Ex
:
50 mph
east
+ 40 mph west = 10 mph eastSlide14
Adding Perpendicular Vectors
When vectors are perpendicular:sketch the vectors in a HEAD TO TAIL orientation use right triangle trig to solve for the resultant and direction.Ex
:
50 mph east
+
40 mph south = ??Slide15
An Airplane flies
north with an airspeed of 650 mph. If the wind is blowing
east
at
50
mph, what is the speed of the plane
as measured from the ground
?Slide16
Adding Perpendicular Vectors
R
θ
50 mph
650 mphSlide17
Examples
Ex1: Find the sum of the forces of 30 lb south and 60 lb east.Ex2: What is the ground speed of a speed boat crossing a river of 5mph current if the boat can move 20mph in still water?Slide18
An airplane flies north with an airspeed of 575 mph.
1. If the wind is blowing 30° north of east
at 50 mph, what is the speed of the
plane as measured from the ground?
2. What if the wind blew south of west?Slide19
Adding Skew
VectorsWhen vectors are not collinear and not perpendicular, we must resort to drawing a scale diagram.Choose a scale and a indicate a compassDraw the vectors Head to Tail
Draw the resultant
Measure the resultant and the angle!
Ex
: 50 mph east + 40 mph south = ??Slide20
Adding Skew Vectors
Measure R with a ruler and measure
θ
with a protractor.
R
θSlide21
Vector Components
Vectors are described using their components.Components of a vector are 2 perpendicular vectors that would add together to yield the original vector.Components are notated using
subscripts.
F
Fx
FySlide22
Adding Vectors by Components
A
BSlide23
Adding Vectors by Components
A
B
Transform vectors so they are
head-to-tail.Slide24
Adding Vectors by Components
A
B
B
y
B
x
A
x
A
y
Draw components of each vector...Slide25
Adding Vectors by Components
A
B
B
y
B
x
A
x
A
y
Add components as collinear vectors!Slide26
Adding Vectors by Components
A
B
B
y
B
x
A
x
A
y
Draw resultants in each direction...
R
y
R
xSlide27
Adding Vectors by Components
A
B
Combine components of answer using the head to tail method...
R
y
R
x
R
qSlide28
Adding Vectors Graphically
When you have many vectors, just keep repeating the process until all are includedThe resultant is still drawn from the origin of the first vector to the end of the last vectorSlide29
Adding Vectors Graphically, final
Example: A car travels 3 km North, then 2 km Northeast, then 4 km West, and finally 3 km Southeast. What is the resultant displacement?
A
B
C
D
R
R
is ~2.4 km, 13.5
°
W of N
or 103.5
º
from +
ve
x-axis.
D
C
A
BSlide30
Components of a Vector
A car travels 3 km North, then 2 km Northeast, then 4 km West, and finally 3 km Southeast. What is the resultant displacement? Use the component method of vector addition.
A
B
C
D
B
x
B
y
D
y
D
x
A
x
= 0 km
B
x
= (2 km)
cos
45
º
= 1.4 km
C
x
= -4 km
D
x
= (3km)
cos
45
º
= 2.1 km
X-components
Y-components
A
y
= 3 km
B
y
= (2 km) sin 45
º
= 1.4 km
C
y
= 0 km
D
y
= (3km) sin 315
º
= -2.1 km
x
y
N
S
W
ESlide31
Components of a Vector
S
R
x
= A
x + B
x
+
C
x
+
D
x
=
0 km + 1.4 km - 4.0 km + 2.1 km = -0.5 km
R
y
= A
y
+ B
y
+ C
y
+
D
y
= 3.0 km + 1.4 km + 0 km - 2.1 km = 2.3 km
E
R
x
y
N
W
R
x
R
y
Magnitude:
Direction:
Stop. Think. Is this reasonable?
NO!
Off by 180
º.
Answer: -78º + 180° = 102°Slide32
Adding Vectors by ComponentsSlide33
Components of a Vector
The x-component of a vector is the projection along the x-axis:The y-component of a vector is the projection along the y-axis:Slide34
Adding Vectors by ComponentsUse the Pythagorean Theorem and Right Triangle Trig to solve for R and θ…Slide35
Units
Quantity . . . Unit (symbol) Displacement & Distance . . . meter (m)Time . . . second (s)Velocity & Speed . . . (m/s)Acceleration . . . (m/s2)Mass . . . kilogram (kg)
Momentum . . . (kg
·
m/s)Force . . .Newton (N)Energy . . . Joule (J)
Units are not the same as quantities!Slide36
SI Prefixes
Little Guys
Big GuysSlide37
Kinematics definitions
Kinematics – branch of physics; study of motionPosition (x) – where you are locatedDistance (d ) – how far you have traveled, regardless of direction Displacement (
x
)
– where you are in relation to where you startedSlide38
Distance vs. Displacement
You drive the path, and your odometer goes up by 8 miles (your distance).Your displacement is the shorter directed distance from start to stop (green arrow).What if you drove in a circle?
start
stopSlide39
Speed, Velocity, & Acceleration
Speed (v) – how fast you go Velocity (v) – how fast and which way;
the rate at which position changes
Average speed (
v
) – distance / time Acceleration (
a
) – how fast you speed
up, slow down, or change direction;
the rate at which velocity changesSlide40
Speed vs. Velocity
Speed is a scalar (how fast something is moving regardless of its direction). Ex: v = 20 mphSpeed is the magnitude of velocity.Velocity is a combination of speed and direction. Ex:
v
= 20 mph at 15
south of westThe symbol for speed is v.The symbol for velocity is type written in bold: v
or hand written with an arrow:
vSlide41
Speed vs. Velocity
During your 8 mi. trip, which took 15 min., your speedometer displays your instantaneous speed, which varies throughout the trip.Your average speed is 32 mi/hr.Your average velocity is 32 mi/hr in a SE direction.At any point in time, your velocity vector points tangent to your path.
The faster you go, the longer your velocity vector.Slide42
Acceleration
Acceleration – how fast you speed up, slow down, or change direction; it’s the rate at which velocity changes. Two examples:
t
(s)
v
(mph)
0
55
1
57
2
59
3
61
t
(s)
v
(m/s)
0
34
1
31
2
28
3
25
a
= +2 mph
/
s
a
= -3
m
/
s
s
= -3 m
/
s
2Slide43
Velocity & Acceleration Sign Chart
V E L O C I T Y
ACCELERATION
+
-
+
Moving forward;
Speeding up
Moving backward;
Slowing down
-
Moving forward;
Slowing down
Moving backward;
Speeding upSlide44
Acceleration due to Gravity
9.8 m/s
2
Near the surface of the Earth, all objects accelerate at the same rate (ignoring air resistance).
a
= -
g
= -9.8 m/s
2
Interpretation
: Velocity decreases by 9.8 m/s each second, meaning velocity is becoming less positive or more negative. Less positive means slowing down while going up. More negative means speeding up while going down.
This acceleration vector is the same on the way up, at the top, and on the way down!Slide45
Kinematics Formula Summary
(derivations to follow)
v
f
= v
0
+
a
t
v
avg
= (
v
0
+
v
f
)
/
2
x
=
v
0
t
+
½
a
t
2
v
f
2
–
v
0
2
= 2
a
x
For 1-D motion with
constant
acceleration:Slide46
Kinematics Derivations
a = v /
t
(by definition) a
= (
v
f
– v
0
)
/
t
v
f
= v
0
+
a
t
v
avg
= (
v
0
+
v
f
)
/
2
will be proven when we do graphing.
x
=
v
t
= ½ (
v
0
+
v
f
)
t
= ½ (
v
0
+
v
0
+
a
t
)
t
x = v
0
t
+
a
t
2
(cont.)Slide47
Kinematics Derivations
(cont.)
v
f
=
v
0
+
a
t
t
=
(
v
f
– v
0
)
/
a
x
=
v
0
t
+
a
t
2
x =
v
0
[
(
v
f
–
v
0
)
/
a
]
+
a
[
(
v
f
–
v
0
)
/
a
]
2
v
f
2
–
v
0
2
= 2
a
x
Note that the top equation is solved for
t
and that expression for
t
is substituted twice (in red) into the
x
equation. You should work out the algebra to prove the final result on the last line.Slide48
Sample Problems
You’re riding a unicorn at 25 m/s and come to a uniform stop at a red light 20 m away. What’s your acceleration? A brick is dropped from 100 m up. Find its impact velocity and air time.An arrow is shot straight up from a pit 12 m below ground at 38 m/s.
Find its max height above ground.
At what times is it at ground level? Slide49
Multi-step Problems
How fast should you throw a kumquat straight down from 40 m up so that its impact speed would be the same as a mango’s dropped from 60 m?A dune buggy accelerates uniformly at 1.5 m/s2
from rest to 22 m/s. Then the brakes are applied and it stops 2.5 s later. Find the total distance traveled.
19.8 m/s
188.83 m
Answer:
Answer:Slide50
Graphing !
x
t
A
B
C
A … Starts at home (origin) and goes forward slowly
B … Not moving (position remains constant as time progresses)
C … Turns around and goes in the other direction
quickly, passing up home
1 – D MotionSlide51
Graphing w/ Acceleration
xA … Start from rest south of home; increase speed gradually
B …
Pass home; gradually slow to a stop (still moving north)
C …
Turn around; gradually speed back up again heading southD … Continue heading south; gradually slow to a stop near the starting point
t
A
B
C
DSlide52
Tangent Lines
t
SLOPE
VELOCITY
Positive
Positive
Negative
Negative
Zero
Zero
SLOPE
SPEED
Steep
Fast
Gentle
Slow
Flat
Zero
x
On a position vs. time graph:Slide53
Increasing & Decreasing
t
x
Increasing
Decreasing
On a position vs. time graph:
Increasing
means moving forward (positive direction).
Decreasing
means moving backwards (negative direction).Slide54
Concavity
t
x
On a position vs. time graph:
Concave up
means positive acceleration.
Concave down
means negative acceleration.Slide55
Special Points
t
x
P
Q
R
Inflection Pt.
P, R
Change of concavity
Peak or Valley
Q
Turning point
Time Axis Intercept
P, S
Times when you are at “home”
SSlide56
Curve Summary
t
x
A
B
C
DSlide57
All 3 Graphs
t
x
v
t
a
tSlide58
Graphing Tips
Line up the graphs vertically.
Draw vertical dashed lines at special points except intercepts.
Map the slopes of the position graph onto the velocity graph.
A red peak or valley means a blue time intercept.
t
x
v
tSlide59
Graphing Tips
The same rules apply in making an acceleration graph from a velocity graph. Just graph the slopes! Note: a positive constant slope in blue means a positive constant green segment. The steeper the blue slope, the farther the green segment is from the time axis.
a
t
v
tSlide60
Real life
Note how the v graph is pointy and the a graph skips. In real life, the blue points would be smooth curves and the green segments would be connected. In our class, however, we’ll mainly deal with constant acceleration.
a
t
v
tSlide61
Area under a velocity graph
v
t
“forward area”
“backward area”
Area above the time axis = forward (positive) displacement.
Area below the time axis = backward (negative) displacement.
Net area (above - below) = net displacement.
Total area (above + below) = total distance traveled.Slide62
Area
The areas above and below are about equal, so even though a significant distance may have been covered, the displacement is about zero, meaning the stopping point was near the starting point. The position graph shows this too.
v
t
“forward area”
“backward area”
t
xSlide63
Area units
Imagine approximating the area under the curve with very thin rectangles.Each has area of height width.The height is in m/s; width is in seconds.Therefore, area is in meters!
v
(m/s)
t
(s)
12 m/s
0.5 s
12
The rectangles under the time axis have negative
heights, corresponding to negative displacement.Slide64
Graphs of a ball thrown straight up
x
v
a
The ball is thrown from the ground, and it lands on a ledge.
The position graph is parabolic.
The ball peaks at the parabola’s vertex.
The
v
graph has a slope of -9.8 m/s
2
.
Map out the slopes!
There is more “positive area” than negative on the
v
graph.
t
t
tSlide65
Graph Practice
Try making all three graphs for the following scenario:1. Schmedrick starts out north of home. At time zero he’s driving a cement mixer south very fast at a constant speed.
2. He accidentally runs over an innocent moose crossing the road, so he slows to a stop to check on the poor moose.
3. He pauses for a while until he determines the moose is squashed flat and deader than a doornail.
4. Fleeing the scene of the crime, Schmedrick takes off again in the same direction, speeding up quickly.
5. When his conscience gets the better of him, he slows, turns around, and returns to the crash site.Slide66
Kinematics Practice
A catcher catches a 90 mph fast ball. His glove compresses 4.5 cm. How long does it take to come to a complete stop? Be mindful of your units!
2.24 ms
AnswerSlide67
Uniform Acceleration
When object starts from rest and undergoes constant acceleration:Position is proportional to the square of time.Position changes result in the sequence of odd numbers.Falling bodies exhibit this type of motion (since g is constant).
t
: 0 1 2 3 4
x
= 1
x
= 3
x
= 5
( arbitrary units )
x
: 0 1 4 9 16
x
= 7Slide68
Spreadsheet Problem
We’re analyzing position as a function of time, initial velocity, and constant acceleration.x, x, and the ratio depend on t
,
v
0
, and a
.
x
is how much position changes each second.
The ratio (1, 3, 5, 7) is the ratio of the
x
’s.
Make a spreadsheet like this and determine what must be true about
v
0
and/or
a
in order to get this ratio of odd numbers.
Explain your answer mathematically.Slide69
Relationships
Let’s use the kinematics equations to answer these:1. A mango is dropped from a height h.
a. If dropped from a height of 2
h
, would the impact speed double?Would the air time double when dropped from a height of 2 h
?
A mango is thrown down at a speed
v
.
If thrown down at 2
v
from the same height, would the impact speed double?
Would the air time double in this case?Slide70
Relationships (cont.)
A rubber chicken is launched straight up at speed v from ground level. Find each of the following if the launch speed is tripled (in terms of any constants and
v
).
max height
hang time impact speed
3
v
9
v
2
/
2
g
6
v
/
g
Answers