Post on 26-Mar-2021
transcript
Four Types of Motion We’ll Study
The branch of mechanics that studies the motion of a body without caring about what caused the motion.
Kinematics definitions
Kinematics – branch of physics; study of motion
Position (x) – where you are located
Distance (d ) – how far you have traveled, regardless of direction
Displacement (x) – where you are in relation to where you started
Vector versus ScalarA scalar has only a size - no direction
Example – 20 m/s
A vector has a size (magnitude) and a direction
Example – 20 m/s due North
MotionMotion is just a change in
position over time.
Measurements needed
Distance
Time
Direction helps
Position Mark a zero point on the line, pick a direction to be
positive, and measure from there.
Positions can be positive or negative.
Units of position: centimeters, meters, kilometers, inches, feet, miles, etc.
Common symbol: x
Positions are Relative Different people can mark the line differently, so they
can get different numbers for position.
The position number (and unit) really don’t mean anything until you specify where you marked “0”, and which way you made positive - your frame of reference.
Distance• The total length of the path traveled by an object.
• Does not depend upon direction. (scalar)
• “How far have you walked?”
Displacement• The change in position of an object.
• Depends only on the initial and final positions, not on path.
• Includes direction. (vector)
• “How far are you from home?”
A
B
50 mdisplacement
100 m
distance
Distance vs Displacement
DisplacementRepresented by d
∆d = df – di
-OR-
Represented by x.
x = x2 - x1
where
x2 = final position
x1= initial position
If displacement is positive, the object moves to the right.
If the displacement is negative, the object moves to the left.
Checking Understanding
Maria is at position x = 23 m. She then undergoes a displacement ∆x = –50 m. What is her final position?
A. –27 m
B. –50 m
C. 23 m
D. 73 m
Answer
Maria is at position x = 23 m. She then undergoes a
displacement ∆x = –50 m. What is her final position?
A. –27 m
B. –50 m
C. 23 m
D. 73 m
Rates A rate measures how fast something changes.
In physics, a rate is almost always calculated as a quantity divided by time.
Rate Q changes = change in Q
time for Q to change
SPEED Speed is the rate position changes, or the rate distance is
covered.
Speed is telling how fast something is moving.
Speed is always a positive number. (Remember the distance vs. time graph)
Quantities that can be described by a single number (magnitude) are called scalars.
Measuring Speed Speed is the amount of
distance an object travels in a certain amount of time.
The equation for speed is…
Speed = distance
time
Here is a little bug located at 0 cm at 0 seconds.
10 seconds later he is at 50 cm
SPEEDThe speed of the bug is just his
distance traveled divided by how long it takes.
Average speed = distance/time
Speed = 50 cm/10s
s = 5 cm/s
The skier has traveled 400 meters in 6 seconds. What is his
speed?
Distance traveled is
The time it took was 6 seconds.
The speed is the distance traveled divided by the time taken.
speed = 400 meters
6 seconds
speed = 66.67 ms
Remember: We
have a specific
5-step method
for solving
problems…the
GUESS method.
Problems
When Evelyn Ashford was in the Olympics she broke
the record for the 200 m run by completing it in 11s.
What was her speed?
G: givens
v = 8.1cm/year
t = 1 century = 100 years
U: unknowns
d = ?
E: equation
v = d/t
Rearrange by multiplying sides by t
v(t) = d (t)
t
vt =d or d = vt
S: substitution
d = ( 8.1 cm )(100 yrs )
yr
S: solution
v = 810 cm, or 8.1 m
Problems The Pacific Plate moves at an pace of 8.1 cm/year. How far
does the plate move in a century?
G: givens d = 30.5 m
v = 73.14 m/s
U: unknowns t = ?
E: equation v = d/t
S: substitution
t(v) = d
v v
t = d/v
t = 30.5m
73.14m/s
S: solution t = 0.42 sec
Problems
In 1931, "'Big Bill' Tilden delivered the fastest serve ever officially
measured. The speed was 73.14 m/s. If the serve covered 30.5 m, how
much time did Bill’s opponent have to react before the ball reached him?
VELOCITY
Velocity is speed and direction.
Velocity is how fast and which way.
Quantities that have direction are called vectors.
Average velocity = displacement / time
Units are meters/second (m/s)
Average velocity does not tell you speed or velocity at each moment.
Can be positive or negative depending on direction moved.
Time can never be negative!
Average -VS- Instantaneous Speed or Velocity
Average
• distance / time
•Easier to calculate
Instantaneous
•No easy calculation
•Moment in time or
any given instant
•What's on a cars
speedometer
•On graph study
small time interval
(slope at that point)
Suppose our bug stars off at 100 cm.
And ends up at 80 cm 10 seconds later.
Velocity = change in position/time
Velocity = (final position – initial position)/time
Velocity = (80cm – 100cm)/10 seconds
Velocity = - 2 cm/s
A positive velocity means moving to the right. (usually)
A negative means moving to the left. (usually)
Practice
Heather and Matthew walk eastward with a speed of .98 m/s. If it takes them 34 min to walk to the store, how far have they walked?
Practice Heather and Matthew walk eastward with a speed of .98 m/s.
If it takes them 34 min to walk to the store, how far have they walked?
Knowns? What do you know? Write it down.
Speed = .98 m/s, time = 34 minutes (2040 sec)
Unknown? What do you want to know?
How far? Distance = ?
Equation? Write the equation you’ll use.
Speed = distance / time
Work the problem.
.98 m/s = distance / 2040 sec; d = 1999.2 meters
Position–Time Graphs
Position–Time Graphs
Position of Moving
Object over Time
0
2
4
6
8
10
12
0 5 10 15 20 25 30
Time
Po
sit
ion
• Position is not changing over time.
• Not Moving
• v =0
Position–Time Graphs
Position of Moving
Object over Time
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30
Time
Po
sit
ion
v =Δx = slope
Δt
•Position is changing, i.e. object is moving
•Velocity determined by slope of position-time graph
•For Linear Relationship (straight line), slope is constant.
Position–Time Graphs
Position of Moving
Object over Time
0
10
20
30
40
50
60
0 5 10 15 20 25 30
Time
Po
sit
ion
Position–Time Graphs
Position of Moving
Object over Time
0
20
40
60
80
100
0 5 10 15 20 25 30
Time
Po
sit
ion
•Line A and Line B have same slopes; Object A and Object B moving with same velocity
•Line A and Line B have different y-intercepts; Object A and Object B have different starting
points.
Position–Time Graphs
Position of Moving
Object over Time
0
10
20
30
40
50
60
70
80
0 5 10 15 20 25 30
Time
Po
sit
ion
• Slope is constant, velocity is constant
• Slope is negative, velocity is negative
What does this
mean?
Graph of Average vs. Instantaneous VelocityRecall…
Instantaneous Velocity is velocity at a given instant in time. You can use slope to figure this out…
Position vs. Time
0
5
10
15
20
25
30
35
40
0 5 10 15 20
Time (s)
Posit
ion
(m
)
Vavg= Δd
Δt
Graph of Average and Instantaneous VelocitySlope Remains constant Velocity is constant
Graph of Average and Instantaneous Velocity(curved is not constant velocity)Slope is not constant and changes so Velocity is not constant
Position vs Time
0
5
10
15
20
25
30
35
0 5 10 15 20 25
Time (s)
Po
sit
ion
(m
)
•Velocity is the slope of the tangent line at this point.
Motion Diagrams
Making a Motion Diagram
Examples of Motion Diagrams
Motion Diagrams
Motion Diagrams
The Particle Model
A simplifying model in which we treat the object as if all its mass were concentrated at a single point. This model helps us concentrate on the overall motion of the object.
Position and Time
The position of an object is located along a coordinate system.
At each time t, the object is at some particular position. We are free to choose the origin of time (i.e., when t = 0).
Slide 1-17
Motion Diagrams-Particle a.k.a Oil Drip
Checking Understanding
Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner
is moving faster?
Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner
is moving faster?
Answer
A
Checking Understanding
Two runners jog along a track. The times at each position are shown. Which runner is
moving faster?
C.They are both moving at the same speed.
Two runners jog along a track. The times at each position are shown. Which runner is
moving faster?
C.They are both moving at the same speed.
Answer
ACCELERATION Acceleration is a change in velocity per time.
a = (vf-vi)/t a = Δv/t
So an object accelerates if it…
Speeds up
Slows down
Or changes the direction it is moving,
Acceleration = change in velocity/time
Acceleration = (final velocity – initial velocity)/time
Acceleration has units of …
-velocity/time
-(meters per second)/seconds
-Or m/s/s or m/s2
General RuleIf the sign of the velocity and the
sign of the acceleration is the same, the object speeds up.
If the sign of the velocity and the sign of the acceleration are different, the object slows down.
Acceleration is a vector because it has size and direction.
Ex.
If the final velocity is greater than the initial velocity the ∆v will be positive
If the final velocity is less than the initial velocity the ∆v will be negative .
confusion Acceleration, like velocity, has direction.
For objects moving along a straight line the direction of an object’s acceleration is denoted by plus or minus.
Accelerating vs Decelerating
A negative acceleration doesn’t always mean the object is slowing down. It could be moving in the negative direction.
Constant AccelerationVelocity increases or decreases by exactly the same amount during each time interval.
The displacement for each time interval increases or decreases by increasing or decreasing amounts.
5 Kinematic EquationsAll the kinematic equations are related.
They can be derived using the idea that acceleration will be constant.
There is always more than one way to solve each problem.
In general though, one equation is generally easier to use then others.
KINEMATIC EQUATIONS Let’s review the quantities we’ve seen so far:
The fundamental quantities are displacement (x or y), velocity (v), and acceleration (a).
Acceleration is a change in velocity, from an initial velocity (v0 or vi) to a final velocity (v or vf).
KINEMATIC EQUATIONS And finally, the motion takes place during some
elapsed time interval, Δt.
If we agree to start our clocks at time ti = 0, then we can just write t instead of Δt, which simplifies the notation.
Therefore, we have five kinematic quantities:
x, v0, v, a, and t.
Equations - Kinematica = (vf-vi)/ (tf-ti) OR ∆v/∆t
vavg = ½ (vi +vf)
Now using substitutions….
∆d = 1/2(vi +vf)∆t
vf = vi +a∆t
df =di + vi(tf) + ½ a(tf)2
vf2 = vi
2 +2a∆d
KINEMATIC EQUATIONS These five quantities are related by a group of equations
that we call the BIG FOUR:
Variable missing
BIG FOUR #1: x = ½(vi + vf)t a
BIG FOUR #2: vf = vi + at x
BIG FOUR #3: xf = xi + vit + ½at2 vf
BIG FOUR #4: vf2 = vi
2 + 2ax t
KINEMATICS BIG FOUR Each of the BIG FOUR equations is missing one
of the five fundamental quantities.
The way you decide which of equation to use when solving a problem is to determine which of the fundamental quantities is missing from the problem – that is, which quantity is neither given nor asked for – and then use the equation that doesn’t have that variable.
KINEMATICS BIG FOUR For example, if the problem never mentions the final
velocity …
… v is neither given nor asked for …
… the equation to use is the one that’s missing vf …
…that’s BIG FOUR #3 …
xf = xi + vit + ½at2
The equations Tips
It can be confusing, but sometimes
These equations are written
different.
In this case “x” is distance or “d”.
Also, the subscripts “i” and “f”
have
been replaced with “o” and no
subscript for “f”.
Tips You won’t get a problem wrong by using the wrong
equation. You’ll get no answer at all because a piece of information is missing.
The first step is to READ CAREFULLY!
Tips Read slowly, carefully, and more than once.
For example, you might gloss over that an “object starts at rest” and later be confused as to the missing initial velocity.
However, when an object “starts at rest” it is implied that the initial velocity is 0m/s.
tips Know what you’re looking for.
“What is the final velocity?” means V=?
“How far will the car travel before it comes to a stop?” means d=?
Tips Choose the correct equation.
Choose the equation that best relates to the facts given.
If the problem doesn’t include time, you probably won’t be using an equation using time.
Practice Problems…Do on BoardA hockey player glides along the ice at a
constant speed of 1.25 m/s in the positive direction onto a rough section of ice, which slows him. If he stops in 5.0 s, what is the magnitude and direction of his acceleration?
a = (vf-vi)/ (tf-ti) OR ∆v/∆t
a = o.25 m/s2 , negative
Practice Problems…Do on Board
A race car travels on a racetrack at 44 m/s and slows at a constant rate to a velocity of 22 m/s over 11 s. How far does it move during this time?
∆d = ½ (vi +vf)∆t
363 m
Practice Problems…Do on BoardJill jogs at a velocity of 2.50 m/s. If she
then accelerates at a constant -0.10 m/s2, how fast will she be jogging when she has moved 10.0m ?
vf2 = vi
2 +2a∆d
21 m/s
Acceleration Practice
If a car can go from 0 mi/hr to 60 mi/hr in 8 seconds,
what is its acceleration?
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Acceleration Practice
A car must come to an emergency stop. What is its acceleration if it took
8 seconds for it to stop when it was traveling at 30 m/s?
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Velocity –Time Graphs
Velocity-Time Graphs Have described motion using Position-Time Graphs.
Can also describe motion using Velocity-Time Graphs.
Graphs
The slope and shape of a graph describes the object’s motion.
Velocity-Time GraphsConstant Velocity
Velocity of Moving
Object over Time
0
2
4
6
8
10
12
0 5 10 15 20 25 30
Time
Velo
cit
y (
m/
s)
• Δd= vΔt
• Displacement = Area of under curve
(rectangle)
Velocity-Time GraphsVelocity is Not Constant
Velocity of Moving
Object over Time
0
1
2
3
4
5
6
7
0 5 10 15 20 25 30
Time
Velo
cit
y (
m/
s)
• Velocity increasing over time
• Δd= Area of under curve (triangle)
Velocity-Time GraphsVelocity is Not Constant
Velocity of Moving
Object over Time
0
1
2
3
4
5
6
7
8
9
0 5 10 15 20 25 30
Time
Velo
cit
y (
m/
s)
• Velocity decreasing over time
• Δd= Area of under curve (triangle)
Velocity-Time GraphsVelocity is Not Constant
Velocity of Moving
Object over Time
0
1
2
3
4
5
6
7
0 10 20 30 40 50 60 70 80 90 100
Time
Velo
cit
y (
m/
s)
How Object Moving?
• From t0 to t30: Velocity increasing
• From t30 to t60: Velocity constant
• From t60 to t90: Velocity decreasing
Pick the constant velocity graph(s)…
A
t
x
C
t
v
B
t
x
D
t
v
Here is a motion diagram of a car moving along a straight stretch of road:
Which of the following velocity-versus-time graphs matches this motion diagram?
Checking Understanding
A. B. C. D.
Here is a motion diagram of a car moving along a straight stretch of road:
Which of the following velocity-versus-time graphs matches this motion diagram?
Answer
A. B. C. D.
Checking Understanding
A graph of position versus time for a basketball player moving down the court appears like so:
Which of the following velocity graphs matches the above position graph?
A. B. C. D.
A graph of position versus time for a basketball player moving down the court appears like so:
Which of the following velocity graphs matches the above position graph?
Answer
A. B. C. D.
A graph of velocity versus time for a hockey puck shot into a goal appears like so:
Which of the following position graphs matches the above velocity graph?
Checking Understanding
A. B. C. D.
A graph of velocity versus time for a hockey puck shot into a goal appears like so:
Which of the following position graphs matches the above velocity graph?
Answer
A. B. C. D.
Free Fall Free fall is motion under the influence of gravity
only - no friction or air resistance.
Gravity accelerates the object toward the earth.
Acceleration in Free Fall The acceleration of an object in free fall is constant.
At the surface of Earth, the free-fall acceleration is about 10 m/s2, or 9.8 m/s2 if you have a calculator (or 32 ft/s2 or 22 mi/hr/s in “English” units).
g = 9.8 m/s2 downward.
a = -g if up is positive.
acceleration is down when ball is thrown up EVERYWHERE in the balls flight.
Summary
v = vo - gt
x = xo + vot - 1/2 gt2
v2 = vo2 – 2g(∆x)
Symmetry When something is thrown upward and returns to
the thrower, this is very symmetric.
The object spends half its time traveling up; half traveling down.
Velocity when it returns to the ground is the opposite of the velocity it was thrown upward with.
Acceleration is –9.8 m/s2 everywhere!
Air Resistance The effect of air resistance is to slow an object down
and/or decrease its acceleration.
Gravity ProblemsAt what speed would a penny hit the water in a wishing well if it falls for 3.2 seconds?
G: (For v1: If I drop something what would its initial velocity be, when it is still in my hands?)
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Gravity ProblemsIf a water balloon is dropped out of a 2 story window and hits the person below at 14 m/s, how long was it falling?
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