Physics 1710 Chapter 2 Motion in One Dimension

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Physics 1710Chapter 2 Motion in One
DimensionII

• Galileo Galilei Linceo
• (1564-1642)
• Discourses and Mathematical Demonstrations
  Concerning the Two New Sciences (1638) at age 74!
  (four years before his death at 78)

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Physics 1710Chapter 2 Motion in One Dimension
Galileos Ramp
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Physics 1710Chapter 2 Motion in One
DimensionII

• 1' Lecture
• Under uniform acceleration ax
• Acceleration is constant (hence uniform) ax
  constant (gt0,0, or lt0)
• Velocity changes linearly in time vfinal
  vinitial ax t
• Displacement increases quadratically with
  time xfinal xinitialvinitial t ½ ax t 2
  

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Physics 1710Chapter 2 Motion in One
DimensionII

• Displacement is the change in position.
• ?x xfinal - xinitial

• ? is the change operator
• Change in x x at end x at start
• ?Balance Balancefinal - Balanceinitial

REVIEW
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Physics 1710Chapter 2 Motion in One
DimensionII

• Velocity is the time rate of displacement.
• Average velocity vx, ave ? x / ?t
• Instantaneous velocity vx lim ?t?8 ?x /
  ?t dx/dt

REVIEW
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Physics 1710Chapter 2 Motion in One
DimensionII

Position (m)
vaverage ?x/?t
Time (sec)
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Physics 1710Chapter 2 Motion in One
DimensionII

• Velocity

Plot it!
vx dx/dt Instantaneous Velocity
Position (m)
vx, ave ?x / ?t AverageVelocity
Time (sec)
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Physics 1710Chapter 2 Motion in One
DimensionII

• Acceleration is the time rate of change of
  velocity.
• Average acceleration ax, ave ?vx / ?t
• (vx,final -vx, initial )/ ?t
• Instantaneous acceleration ax lim ?t?8
  ?vx / ?t
• dvx /dt
• ax dvx /dt d(dxx /dt)/dt d2x/dt 2

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Physics 1710Chapter 2 Motion in One
DimensionII

• Motion Map
• From snap shots of motion at equal intervals of
  time we can determine the displacement, the
  average velocity and the average acceleration in
  each case.

Uniform motion
Accelerated motion
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Physics 1710Chapter 2 Motion in One Dimension

• Velocity

Plot them!
a2gt0
a1 0
Position (m)
Velocity (m/sec)
ax dvx /dt
Time (sec)
Time (sec)
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Physics 1710Chapter 2 Motion in One
DimensionII

• Galileo Galilei Linceo

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Physics 1710Chapter 2 Motion in One
DimensionII

• Galileos rule of odd numbers
• Under uniform acceleration from rest,
• a body will traverse distances in successive
  equal intervals of time
• that stand in ratio
• as the odd numbers 1,3,5,7,9

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Physics 1710Chapter 2 Motion in One
DimensionII

• Galileos Ramp Demonstration

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Physics 1710Chapter 2 Motion in One
DimensionII
Observation

• 0 0 0 2
• 01 1 1 2
• 13 4 2 2
• 45 9 3 2
• 97 16 4 2
• 169 25 5 2
• 2511 36 6 2

?x ? t 2 from rest (vinitial 0)
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Physics 1710Chapter 2 Motion in One
DimensionII

• Kinematic Equations from Calculus

(dv/dt) a constant ?0t (dv/dt) dt ?0t a
dt ?vinitialvfinal dv a (t-0) ?v v vinitial
at The change in the instantaneous velocity is
equal to the (constant) acceleration multiplied
by its duration. v vinitial at
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Physics 1710Chapter 2 Motion in One
DimensionII

• Kinematic Equations from Calculus

dx/dt v ?0t (dx/dt) dt ?0t v dt x
xinitial ?0t (vinitial at) dt
vinitial (t-0) ½ a(t 2-0) ?x vinitial t ½
at 2 The displacement under uniform acceleration
is equal to the displacement at constant velocity
plus one half the acceleration multiplied by the
square of its duration. x xinitial vinitial
t ½ at 2
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Physics 1710Chapter 2 Motion in One
DimensionII

• Kinematic Equations from Calculus

v 2 (vinitial at) 2 v 2 vinitial 2 2
vinitial at a 2 t 2 v 2 vinitial 2 2a
(vinitial t ½ at 2) v 2 vinitial 2
2a?x The change in the square of the velocity
is equal to two times the acceleration multiplied
by the distance over which the acceleration is
applied.
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Physics 1710Chapter 2 Motion in One
DimensionII
80/20 facts
Kinematic Equations (1-D Uniform a)

• vx final vx initial ax t
• x final x initial vx initial t ½ axt 2
• vx final 2 vx initial 2 2 ax (x final - x
  initial )

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Physics 1710Chapter 2 Motion in One
DimensionII
80/20 facts

• In free fall near the Earth, all bodies are
  accelerated uniformly downward with an
  acceleration of
• az - g -9.80 m/s2.

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Physics 1710Chapter 2 Motion in One
DimensionII

• Plot it!

Position (m)
Acceleration (m/s/s)
Velocity (m/sec)
0
0
dx/dt ?
dv/dt ?
-9.8 m/s/s
Time (sec)
Time (sec)
Time (sec)
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Physics 1710Chapter 2 Motion in One
DimensionII

• The change in the instantaneous velocity is
  equal to the (constant) acceleration multiplied
  by its duration. ?v at
• The displacement is equal to the displacement at
  constant velocity plus one half of the product of
  the acceleration and the square of its duration.
  ?x vinitial t ½ at 2
• The change in the square of the velocity is equal
  to two times the acceleration multiplied by the
  distance traveled during acceleration. ?v 2
  2a ?x
• The acceleration of falling bodies is 9.8 m/s/s
  downward. a - g - 9.8 m/s/s

• Summary

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Physics 1710Chapter 2 Motion in One
DimensionII

• 1' EssayOne of the following

• The main point of todays lecture.
• A realization I had today.
• A question I have.