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TRANSCRIPT
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Motion in One Dimension
Chapter 3
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Outline
3.1 Position, Velocity and Speed
3.2 Instantaneous Velocity and Speed
3.3 Acceleration
3.4 Motion Diagrams
3.5 One-Dimensional Motion with Constant Acceleration
3.6 Freely Falling Objects
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3.1 Position, Velocity and Speed
Position Displacement
An object’s position is its
location with respect to a
chosen reference point
• the change in position during
some time interval
• represented by x
x ≡ xf – xi
x is positive when xf > xi
x is negative when xf < xi
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Vectors and Scalars
A vector – physical quantity that require the specification of both
direction and magnitude
use “+” and “-” sign to indicate vector directions
ex : displacement, velocity and acceleration
Displacement in positive (+x) means object move to the right and (-
x) means move to left
• A scalar – A quantity that has magnitude but no direction
ex : position, speed and etc.
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3.1 Position, Velocity and Speed cont.
Average speed Average velocity
Speed is a scalar quantity
Defined as
It has no direction, so always positive value.
• the displacement x occurs at the interval of time.
x indicates motion along x- axis
Average velocity of a moving particle can be positive and negative depends on the sign of displacement
• x is positive when xf > xi , Vx. avg is positive
x is negative when xf < xi , , Vx. avg is negative
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3.1 Position, Velocity and Speed cont.
Average speed Average velocity
Distance traveled per unit of time Displacement traveled per unit of time
A
Bd = 30 m
Time t = 5 s
• Direction is not (scalar)
A
Bd = 30 m
Time t = 5 s
x =12 m
20o
V = 2.4 m/s at 200 N of E
•Direction required (vector)
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3.2 Instantaneous Velocity and Speed
Instantaneous velocity can indicate what is happening at
every point of time.
General equation for instantaneous velocity :
Instantaneous velocity can be positive, negative or zero.
Instantaneous speed is the magnitude of the instantaneous
velocity but it has no direction.
0limx
t
x dxv
t dt
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Instantaneous Velocity, graph
The instantaneous velocity is the
slope of the line tangent to the x
vs. t curve.
This would be the green line.
The light blue lines show that as
t gets smaller, they approach the
green line.
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3.3 Acceleration
Acceleration is the velocity changes with time.
The average acceleration is the change in velocity Vx divided by
the time interval t during the change occurred.
“+” and “-” sign can indicate direction.
,x xf xi
x avg
f i
v v va
t t t
Dimension : L/T2
SI : m /s2
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Instantaneous Acceleration the limit of the average acceleration as t approaches 0.
is also equals to the derivative of the velocity with respect to time.
acceleration is the slope of Vx vs t graph.
It has positive and negative value to indicate the direction.
2
20lim x xxt
v dv d xa
t dt dt
With :
Vx = dx/dt
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Instantaneous Acceleration – graph
The slope of the velocity-time graph is the acceleration.
The green line represents the instantaneous acceleration.
The blue line is the average acceleration.
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Ex 3.4 Average and Instaneous
AccelerationThe velocity of a particle moving along the x axis varies in time according to the expression
Vx = ( 40 – 5t2) m/s, where t is in seconds.
a) Find the average acceleration in the time interval t=0 to t=2s
b) Determine acceleration at t=2s
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Notes about Velocity and Acceleration
Negative acceleration does not necessarily mean the object is
slowing down.
If the acceleration and velocity are in the same direction no
matter “+” / “-” , the object is speeding up.
If the acceleration and velocity are in the opposite direction,
the object is slowing down.
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3.4 Motion Diagram
A motion diagram can be formed by imagining the stroboscope photograph of a moving object.
Red arrows represent velocity.
Purple arrows represent acceleration. All purple arrows maintain the same length means acceleration are constant.
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Constant Velocity
The car image are equally spaced.
The car is moving with constant positive velocity. (red arrows
remain the same size)
Acceleration equals to zero.
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Acceleration same direction with Velocity
The car image farther apart as time increases
Acceleration and velocity are in the same direction
Velocity is increasing as the arrows are getting longer
Acceleration is constant as the arrows maintain the same length
Thus, it is positive acceleration and positive velocity
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Acceleration opposite direction with Velocity
The car image become closer as time increases
Acceleration and velocity are in the opposite direction
Velocity is decreasing as the arrows are getting shorter
Acceleration is constant as the arrows maintain the same length
Thus, it is negative acceleration and positive velocity
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3.5 One-dimensional Motion with Constant
Acceleration
Constant acceleration.
Velocity changes at the same rate throughout the motion.
Kinematic Equation 1 :
xf xi xv v a t
Can determine the velocity at any
time t when we know its initial velocity
and its acceleration
Doesn’t tell about displacement
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Kinematic Equation 2:
The average velocity in a time interval can be expressed as the
arithmetic mean of the initial velocity Vxi and final velocity Vxf
is only suit for constant acceleration.
,2
xi xfx avg
v vv
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Kinematic Equation 3:
This gives us about the position of the object in terms of
time and velocities.
Doesn’t tell acceleration
Thus,
,2
xi xfx avg
v vv
Substitute Equation 2
then
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Kinematic Equation 4:
This gives final position in terms of velocity and acceleration.
Doesn’t tell final velocity
Substitute Equation 3 with Equation 1
xf xi xv v a t
then 21
2f i xi xx x v t a t
Equation 3 :
Equation 1:
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Kinematic Equation 5:
Tells final velocity in terms of acceleration and displacement.
Doesn’t include with time
xf xi xv v a t Equation 1: then
Equation 3 :
Substitute Equation 1 with Equation 3
2 2 2xf xi x f iv v a x x
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When a = 0
When the acceleration is zero,
vxf = vxi = vx xf = xi + vx t
When acceleration is zero, velocity is constant and displacement
changes linearly with time.
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Kinematic Equations - Summary
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Example (3.7) : Carrier Landing
A jet lands on an aircraft carrier at 140 mi/h (63 m/s).
(a) What is its acceleration if it stops in 2.0 s?
(b) What is the displacement of the plane while it is stopping?
Example (3.8) : Watch Out for the Speed Limit!
A car traveling at a constant speed of 45.0 m/s passes a trooper hidden behind
a billboard. One second after the speeding car passes the billboard, the
trooper sets out from the billboard to catch it, accelerating at a constant
rate of 3.00 m/s2. How long does it take her to overtake the car?
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3.6 Freely Falling Objects
A freely falling objects is any object that moves freely under
the influence of gravity alone
It doesn’t depend upon the initial motion of the object.
Thrown upward
Thrown downward
Dropped (release from rest)
• Any freely falling object experiences an acceleration directed
downward, regardless of the initial motion.
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3.6 Freely Falling Objects cont.
The magnitude of free fall acceleration is symbol g
g decreases with increasing altitude
g varies with latitude
Define “up” as +y direction
the g value at Earth’s surface is 9.80 m/s2
• If we neglect air resistance, assume acceleration doesn’t vary with
altitude over distance, the free fall motion is constantly accelerated
motion in one dimension.
• While the motion in vertical direction and acceleration is downward
• we always take ay = -g = - 9.80 m/s2, the minus sign shows the acceleration of a
freely falling object is downward.
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Free Fall – An Object Dropped
•Initial velocity, vo = 0
•Assume upward direction is positive
•Use the kinematic equations
– Generally use y instead of x since vertical
•Acceleration is
– ay = -g = -9.80 m/s2
vo= 0
a = -g
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Free Fall – An Object Thrown Downward
•ay = -g = -9.80 m/s2
•Initial velocity 0
– With upward being
positive, initial velocity
will be negative.
vo≠ 0
a = -g
Section 2.7
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Free Fall – Object Thrown Upward
•Initial velocity is upward, so
positive
•At the maximum height, The
instantaneous velocity is zero.
•ay = -g = -9.80 m/s2
everywhere in the motion
v = 0
vo≠ 0
a = -g
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6.6 Sign Convention:A Ball Thrown Vertically
Upward
•• Velocity is positive (+) Velocity is positive (+) •• Velocity is positive (+) Velocity is positive (+) or negative (or negative (--) based ) based
on on direction of motiondirection of motion..
•• Displacement is positive Displacement is positive ) based ) based
•• Displacement is positive Displacement is positive (+) or negative ((+) or negative (--) based ) based
on on LOCATIONLOCATION. .
Release Point
UP = +
TippensTippens
•• Acceleration is (+) or (Acceleration is (+) or (--) ) based on direction of based on direction of forceforce(weight).(weight).
y = 0
y = +
y = +
y = +
y = 0
y = -
v = +
v = 0
v = -
v = -
v= -
a = -
a = a = --
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Example (3.10) : Not a bad Throw
for a Rookie!
A stone thrown from the top of a
building is given an initial velocity
of 20.0 m/s straight upward. The
building is 50.0 m high, and the
stone just misses the edge of the
roof on its way down, as shown
in Figure (2.14). Using tA = 0 as
the time the stone leaves the
thrower’s hand at position ,
determine :
(a) The time at which the stone
reaches its maximum height,
(b) The maximum height,
(c) The time at which the stone
returns to the height from which it
was thrown,
(d) The velocity of the stone at this
instant, and
(e) The velocity and position of the
stone at t = 5.00 s