September 15, 2009Physics I Lesson 2 Dr J. Tison 1 Kinematics: One Dimensional Motion.
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Transcript of September 15, 2009Physics I Lesson 2 Dr J. Tison 1 Kinematics: One Dimensional Motion.
September 15, 2009 Physics I Lesson 2Dr J. Tison
2
Short History of Classical Mechanics:Greeks – 17th and 18th Century
• Aristotle and other Greek philosophers– Theorized, no experimental data
• Muslim physicists in 11th and 12th centuries– Scientific methods first applied
• Inertia, momentum, force and mass, gravity
– Precursor to Newton’s Laws of Motion
• Modern interpretation of principles of motion– Keppler (-1630)
• Planetary motion observations by Tyco Brahe
– Galileo (-1642)• Acceleration of rolling ball on inclined plane
– Newton (-1727)• Principia… Laws of Motion
September 15, 2009 Physics I Lesson 2Dr J. Tison
3
Classical Mechanics
• Kinematics– How objects move– Equations relating
• Distance, velocity, acceleration, time
• Dynamics– Why things move– Equations relating
• Momentum, force, potential energy, pressure and power.
September 15, 2009 Physics I Lesson 2Dr J. Tison
4
Motion Types
• Translational– 1 Dimension (this lesson)
• Straight line motion
– 2 & 3 Dimension (later lesson)• Trajectory motion
• Rotational (later lessons)
• Combined Translation and Rotation (later lessons)
September 15, 2009 Physics I Lesson 2Dr J. Tison
5
1D Motion:Model and Concepts
• ‘Idealized’ particle– Mathematical point– No spatial extent (no size)
September 15, 2009 Physics I Lesson 2Dr J. Tison
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1D Motion:Frame of Reference
• Measurements (position, distance, speed)– Requires reference point– Observer vs Object
September 15, 2009 Physics I Lesson 2Dr J. Tison
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1D Motion: Frame of Reference
• Coordinate System– Cartesian coordinate system (x,y axes)
• x (horizontal motion); y (vertical motion)
– Position, and distance along x or y axis – Displacement vs distance
• Change in position vs how far object moved
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Motion along a Line• Distance moved along
x axis– ∆x = x2-x1
• Time moved along x axis– ∆t = t2-t1
• Average speed moved along x axis– ∆v = ∆x/∆t
∆ is symbol for ‘change’:final value less
initial value (displacement)
2010 30 x400
y
Distance (m)
S
E
N
W-20 -10 302010-30
Distance =a+b
Displacement =bb-aa
bb
aa
ba
(x1, t1) (x2, t2)
40
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Displacement Vector• Vectors vs Scalars
– Vector symbol• V V (B,I,S)(B,I,S)
– Magnitude/direction
• Business woman walking– Displacement vectors
• aa = 30m east• bb = 60m west• Net displacement vector (cc)
a a + bb = 30m west
– Total distance (not vector)• |b|+ |a| = 90m
S
E
N
W-20 -10 30m2010-30
bb
aa
cc
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Speed & Velocity
• Speed (scalar)– How far object travels in a given time interval
• average speed = (distance traveled)/(time elapsed)
• Velocity (vector)– How far and in what direction traveled in given
time interval• average velocityaverage velocity = (displacementdisplacement)/(time
elapsed)
• Vector = scalar (magnitude) + direction
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Average Speed & Average Velocity:Walking Businesswoman
• Average speed– ∆x(distance)/∆t = [|x2- x1| + |x3- x2|]/|t3- t1|
= (30+60)/(90)
= 1m/s
• Average velocity– ∆xx(displacement)/∆t = (x3- x1)]/(t3- t1)
= (-20-10)/(90)
= -30/90
= -0.33m/s (West)
S
N
W-20 -10 302010-30
bb
aa
40m
tt1=0 at xx1=10m
tt2=30s at xx2= 40m
tt3=90s at xx3=-20m
xx33
xx11 xx22
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Instantaneous Velocity• Constant or changing
velocity• Instantaneous velocity is..
– Velocity at an instant of time
– “Average velocity during an infinitesimal time interval”
• v =lim(∆t →0)[∆x/∆t]
v: instantaneous
vv: average
lim∆t →0
Vel
oci
ty (
mp
h)V
elo
city
(m
ph)
Time (h)Time (h)
2020
00 0.10.1 0.40.40.20.2 0.30.3
4040
6060
Time (h)Time (h)
2020
00 0.10.1 0.40.40.20.2 0.30.3404
0606
0V
elo
city
(m
ph)
Vel
oci
ty (
mp
h)
Average velocityAverage velocity
●● Instantaneous velocityInstantaneous velocity
lim∆t →0
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Acceleration
• Changing velocity = acceleration– Average acceleration = (change of velocity/(time elapsed)
– aa = (v2-v1)/(t2 - t2) = ∆v/∆t
• Instantaneous acceleration– a =lim(∆t →0)[∆v/∆t]
tt1 =0, =0, v1 =0 =0
t =1.0S, t =1.0S, v = = 15mph15mph
t =2s, t =2s, v1 =30mph =30mph t =5s, t =5s, v =60mph =60mph
a =
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Acceleration vs velocity
• Acceleration » speeding up– How quickly velocity changes
• Velocity – How quickly position changes
• Deceleration (a vector) » slowing down – Frame of reference »
• +a to right,
• - a to left
• Acceleration vector is opposite to velocity vector
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Constant Acceleration
• Equations relating x, v, a, and t– Initial conditions
• (t1, x1) = (t0, x0) = (0,0)
• (t2, x2) = (t, x)
• Condition of motion– Magnitude of acceleration is constant– Motion in a straight line.
» aa (instantaneous) = a (average)
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Constant Acceleration
vv = ∆xx/∆t =(x x -xx0)/t
» x» x = xx0+vv ttt
xx
xx0
xx
tt
vv
vv0
vv
tt
∆x = x-x0
∆t = t-t0 = t
t =0t =0
t =0t =0
∆v = v-v0
∆t = t-t0 = t
tt
aa = ∆vv/∆t = (v v -vv0)/t
» v» v = vv0+aa t
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Constant Acceleration
xx = x0+ <v v >t; – where <v v > is average
velocity– From time t = 0 → t = t,
• velocity goes from vv0 → vv
<v v > = (vv0 +vv)/2
vv
vv0
vv
tt
∆v = v-v0
∆t = t-t0 = t
tt
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Constant Acceleration
xx
xx0
xx
tt
∆x = x-x0
∆t = t-t0 = t
t =0t =0
xx = x0+ <vv >t
Substitute:
<v v > = (vv0 +vv)/2
xx = xx0+ [(vv0 +vv)/2]t
x x = x0+ ½(vv0t +vv t)
x x = x0+ ½(vv0t )+½(vv0+aat)(t)
x x = x0+vv0t +½aat2
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Kinematic Equations of Motion For Constant Acceleration
vv = vv0+aa t » t = (vv - vv0)/aa
x x = x0+vv0t +½a a t2
Substitute for t
x x = x0+vv0 (vv - vv0)/aa +½aa[(vv - vv0)/aa]2
Simplify
(vv2 - vv02) = 2aa(x x -- x0)
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Kinematic Equations of Motion For Constant Acceleration
vv = vv0+aa t
x x = x0+vv0t +½a a t2
(vv2 - vv02) = 2aa(x x -- x0)
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Falling Objects:Uniform (constant)Acceleration
Acceleration due to gravity: g = 9.80m/s2 = 32.0ft/s2
• Galileo studied falling objects
– Postulated: All objects fall with same constant acceleration in absence of air or other resistance
– Predicted: Free falling object travels a distance, y, proportional to time squared (t2)
y ~ t2
– Derived: Free fall equation
y y = y0+vv0t +½g g t2
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Falling Objects:Uniform (constant)Acceleration
No air resistance– In a vacuum– Feathers and boulders
fall with same acceleration ~ g
– x ~ t2
= t == t = ∆ ∆yy
September 15, 2009 Physics I Lesson 2Dr J. Tison
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Graphical Analysis of Linear Motion
Linear Equation: (y = mx + b)
• m is slope
• b is y intercept
Eq of Motion for velocity
• x = x0 + vt
Slope: v = ∆x/∆t
Y intercept: x0
xx
tt
∆x = x-x0
∆t = tx0