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PHYSICS 220
Lecture 02
Motion, Forces, and Newton’s Laws
Textbook Sections 2.2 - 2.4
Lecture 2
Purdue University, Physics 220
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Overview
• Last Lecture
–
–
–
–
Units
Scientific Notation
Significant Figures
Motion
• Displacement: Δx = change of position
• Today
– Velocity
• average
• instantaneous
– Acceleration
• average
• Instantaneous
– Newton’s Laws of Motion
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Position and Displacement
• An object’s change in position is its displacement
– Displacement: Δx = xfinal - xinitial
• Average velocity is the displacement per unit
time:
v ave
xfinal − xinitial x2 − x1 Δx
=
=
=
tfinal − tinitial
t2 − t1
Δt
• If an object moves with a constant speed, the
average velocity is constant throughout the
motion
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Velocity (m/s)
• The average velocity is the change in
position (vector) divided by the change
in time.
Δx x f − xi
vav =
=
Δt
t f − ti
along the direction
of displacement
• Instantaneous velocity is the limit of
average velocity as Δt gets small. It is
the slope of the x(t) plot.
Δx
v = lim
Δt →0 Δt
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Velocity
The following plots are x vs t
x
x
t
x
t
x
t
t
• Which plot represents an object at rest?
• Which plot represents an object with a uniform
velocity in the -x direction?
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Exercise
Find the average
velocity for the
object during the
period t=5 s and
t=6 s
x f − xi
1m − 6m
5m
v=
=
=−
= −5m/s
t f − ti
6sec− 5sec
1sec
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Exercise
If the average velocity of a car during a trip along a straight
road is positive, is it possible for the instantaneous velocity
at some time during the trip to be negative?
A - Yes
correct
The car might have reversed for
a little while along the trip
creating a negative
instantaneous velocity at the
point. If the overall
displacement of the car is
positive for that particular time
interval, than the average
velocity is positive as well.
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B - No
If the car is traveling in a
straight path the velocity
will always be positive.
The car needs to travel in
the opposite direction to
get a negative velocity.
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Velocity vs Speed
• Velocity is a vector
– Only depends on the displacement between the initial and final
positions
– Independent of actual paths between the initial and final positions
– The direction of the velocity gives the direction of the motion
• Speed is a scalar
– The magnitude of the velocity is called the speed
– This is the distance traveled per unit of time
– Depends on the length of the actual path between the initial and
final positions
• Remember that speed and velocity are not the same
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Velocity vs Speed
• One-dimensional motion
– Direction of velocity will be parallel to the x-axis
– Will have only one component
• One-, two- or three-dimensional motion
– Velocity may be positive, negative, or zero
– Speed is equal to the magnitude of the velocity
• Speed cannot be negativ
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Acceleration (m/s2)
• The average acceleration is the change in velocity
divided by the change in time. v(t)
Δv v f − vi
aav =
=
Δt t f − ti
Δv
Δt
t
• Instantaneous acceleration is limit of average
acceleration as Δt gets small. It is the slope of the
v(t) plot.
v(t)
Δv
a = lim
Δt →0 Δt
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t
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Acceleration
Is it possible for an object to have a positive
velocity at the same time as it has a negative
acceleration?
A - Yes
“Yes, the object could be moving
B - No
forward but decelerating or
slowing down.”
If the velocity of some object is not zero, can
its acceleration ever be zero?
A - Yes
“An object can have a constant
B - No
velocity, which means that the
acceleration is zero.”
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Graphical Analysis
• To find the velocity graphically
– Find the slope of the line tangent to
the graph at the appropriate times
– For a time interval, find the slope of
the line connecting the two times
• To find the acceleration graphically
– The acceleration is the slope of the
velocity-time graph
– Find the tangent lines at various
locations on the graph
– Sketch an acceleration graph
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Position vs Time Plots
• Gives location at any time
• Displacement is change in position
• Slope gives velocity
x (m)
3
Position at t=3, x(3) = 1
4
Displacement between t=5 and t=1. Δx = -1.0 m
1.0 m - 2.0 m = -1.0 m
t(s)
-3
Average velocity between t=5 and t=1. v = -0.25 m/s
-1 m / 4 s = -0.25 m/s
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Velocity vs Time Plot
• Gives velocity at any time
• Area gives displacement
• Slope gives acceleration
Velocity at t=2, v(2) = 3 m/s
Displacement between t=0 and t=3: Δx = 7.5 m
t=0 to t=1: ½ (3m/s) (1 s) = 1.5 m
t=1 to t=3: (3m/s) (2 s) = 6 m
v (m/s)
3
1.5
6
4
t(s)
-3
Average velocity between t=0 and t=3? v= 7.5 m / 3s = 2.5 m/s
Change in v between t=5 and t=3. Δv = -2 m/s – 3 m/s = -5 m/s
Average acceleration between t=5 and t=3:
a = -5 m/s / (2 s) = -2.5 m/s2
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Acceleration vs Time Plots
• Gives acceleration at any time
• Area gives change in velocity
a (m/s2)
3
Acceleration at t=4, a(4) = -2
6
m/s2
24
Change in v between t=4 and t=1. Δv = +4 m/s
t=1-3: Δv = (3m/s2)(2s) = 6 m/s
t(s)
-3
t=3-4: Δv = (-2m/s2)(1s) = -2 m/s
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Dropped Ball
A ball is dropped from a height of two
meters above the ground.
• Draw vy vs t
3
v
A
-2
3
4 t
3
B
-2
v
3
4 t
3
D
-2
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v
4 t
v
x
v
C
-2
4 t
E
-2
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y
4 t
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Tossed Ball
A ball is tossed from the ground up a height of two
meters above the ground and falls back down.
y
• Draw v vs t
3
v
-2
A
3
4 t
3
B
-2
v
3
4 t
3
D
-2
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v
4 t
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v
C
-2
v
x
4 t
E
-2
4 t
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Problem
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)
B)
C)
D)
Both average acceleration and average velocity are zero.
Average acceleration is zero but average velocity is not zero.
Average velocity is zero but average acceleration is not zero.
Neither average acceleration nor average velocity are zero.
Vave = Δy/Δt = (yf – yi) / (tf – ti)
=0
aave = Δv/Δt = (vf – vi) / (tf – ti)
Not 0 since Vf and Vi are
Lecture 2
not the same!
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Galileo’s Motion Experiments
• Experimented with balls
on an incline
• When the ball was
released from rest, its
velocity varied with time
– As shown in the graph in b
• The acceleration was
constant and positive
– As shown in the graph in c
– The slope of the line in b is
the value of the
acceleration shown in c
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Galileo’s Motion Experiments
• Repeated the experiment by
rolling the ball up the incline
• Give the ball an initial
velocity
• The slope of the velocity-time
graph is negative
• The slope of the v-t graph
was always constant and
depended upon the angle of
the incline
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Galileo’s Motion Experiments
• The acceleration when a ball rolled up a particular
incline was always equal in magnitude, but
opposite in sign, when compared with the
acceleration when the ball rolled down the same
incline
• Reasoned that if the tilt of the incline was exactly
zero, the ball would move with a constant velocity
• Proposed that on a perfectly horizontal ramp, the
ball would roll forever
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Inertia
• The Principle of Inertia
– An object will maintain its state of motion unless it is
acted upon by a force
• The velocity is its state of motion
– Demonstrated by Galileo’s experiments
– Showed that one can have motion without a force
• Broke Aristotle’s link between force and velocity
– Still did not explain how the force is linked to the
motion
• Newton’s Laws provide this link
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Newton’s First Law
Objects at rest remain at rest and objects in motion
remain in motion in a straight line unless acted upon
by an external agent
INERTIA!
- external agents are called Forces
- Forces change the state of
motion of an object
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Force
• Quantifies the “interaction” between two objects
• Four Fundamental Forces
–
–
–
–
Gravitational force
Electromagnetic force
Strong force
Weak force
• Force is a vector
– Has magnitude and direction
– Be careful when you add two forces!
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Newton’s Second Law
The net force on a body is equal to the product of
the mass of the body and the acceleration of the
body
F = ma
1 N = 1 kg x m/s2
- This is a vector equation
- The direction of the net force is the same as the direction
of the acceleration
- In 3 dimensions Fx = max Fy = may Fz = maz
2N
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Directions
• The direction of the
acceleration is always parallel
to the direction of the total
force
• The velocity and the total
force do not need to be in the
same direction
• Example
– Initial velocity is upward
– The total force is downward
– The acceleration is downward
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Inertia and Mass
• Inertia is also a measure of an object’s resistance
to changes in its motion
• This resistance depends on the object’s mass
– The mass of an object is a measure of the amount of
matter it contains
• SI unit of mass is kg
• Mass is an intrinsic property of an object
– It is independent of the object’s location
– It is independent of the object’s velocity or acceleration
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Newton’s Third Law
For every action there is an equal an opposite
reaction
N
Forces in nature come in pairs
T
F
mg
Ff
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F
F
Ff
The object accelerates
IF F > Ff
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Newton’s Third Law
• When one object exerts
a force on a second
object, the second
object exerts a force of
the same magnitude
and opposite direction
on the first object
– Often called the actionreaction principle
• Example
– Force on ball
– Force on bat
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Summary of Concepts
• Velocity: rate of change of position
– average : Δx/Δt
– instantaneous: slope of x vs. t
• Acceleration: rate of change of velocity
– average: Δv/Δt
– instantaneous: slope of v vs. t
• Acceleration and velocity do not necessarily reach a maximum
value at the same time
• The acceleration can be in the opposite direction to the velocity
• Newton’s Laws of Motion
– Inertia
– F = ma
– Pairs
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