acceleration Simplest case a=constant. Equations hold even

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acceleration

• Simplest case aconstant. Equations hold even
  if ?t large.
• ?v vf -vi

ti 0
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Example If a car traveling at 28 m/s is brought
to a full stop 4.0 s after the brakes are
applied, find the average acceleration during
braking.
Given vi 28 m/s, vf 0 m/s, and ?t 4.0 s.
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If a const.
Not true in general
If a const. one dimensional motion
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Fig. 04.01
Vav gives same area Hence same distance
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Constant acceleration
?x vavt
?x vit1/2 at2 ti 0
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Fig. 04.02
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?xvx?tArea
vi?t (blue area)½ ?t (a?t) (gold area)
vi?t ½ a?t2
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aconstant

• ?xvav?t 1/2(vivf)?t
• but vf vi a?t so
• 
  ?t (vf-vi)/a
• ?x 1/2(vivf)?t 1/2 (vivf) (vf-vi)/a
• 1/(2a) (vf2-vi2) ?x
• (vf2-vi2) 2a?x

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Plotted vs time
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Minimum length of runway

• A fully loaded 747 with all engines at full
  throttle accelerates at 2.6 m/s2. Its minimum
  takeoff speed is 70 m/s. How long will it
  require to reach take off speed? What is the
  minimum length of a runway for a 747.

vf vi a?t
(vf2-vi2) 2a?x
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Problem

• A car traveling at a speed of 30 m/s. A deer
  runs across the road and the driver slams on the
  brakes. It takes .75 s to begin applying the
  brakes. With the brakes on the car decelerates
  at 6 m/s2. How far does the car travel from the
  instant the driver sees the deer until he stops.

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Free Fall

• All objects, under the influence of only gravity
  fall. (We are neglecting air resistance)
• They all fall with a constant acceleration (down)
  of
• g 9.8 m/s2
• The mass of the object doesnt matter! Heavy and
  light objects all fall with the same g
• It doesnt matter in which direction it is moving
  it has an acceleration of g
• Since we normally take y up free fall is -g

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Free Fall
Slide 2-36
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• You drop a stone off a cliff and hear it hit the
  ground after two seconds. How high is the cliff?
• This is an how far question
• ?x1/2 at2
• Substitute the numbers a9.8 t2
• ?x19.6
• How fast is it going when it hits?

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Example Problem
Tennis balls are tested by measuring their bounce
when dropped from a height of approximately 2.5
m. What is the final speed of a ball dropped from
this height?
Slide 2-34
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Throwing stones (up)

• What happens if you toss a stone straight up?
  
• v(3)
• v(1)
• v(0)

. v(4) 0. It reaches its highest point when
it stops going up, i.e. when v 0
a is always downward it is g
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Throwing stones (up)

• What happens when it starts coming down again?
  
• v(3)
• v(1)
• v(0)

. v(4) 0. It reaches its highest point when
it stops going up and then begins to
fall with a-g. It re-traces its path
and velocity but down
a is always downward it is g
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Checking Understanding
An arrow is launched vertically upward. It moves
straight up to a maximum height, then falls to
the ground. The trajectory of the arrow is noted.
Which choice below best represents the arrows
acceleration at the different points?

• A ? E ? B ? D C ? 0
• E ? D ? C ? B ? A
• A ? B ? C ? D ? E
• A ? B ? D ? E C ? 0

Slide 2-37
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Answer
An arrow is launched vertically upward. It moves
straight up to a maximum height, then falls to
the ground. The trajectory of the arrow is noted.
Which choice below best represents the arrows
acceleration at the different points?

• A ? E ? B ? D C ? 0
• E ? D ? C ? B ? A
• A ? B ? C ? D ? E
• A ? B ? D ? E C ? 0

Slide 2-38
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Checking Understanding
An arrow is launched vertically upward. It moves
straight up to a maximum height, then falls to
the ground. The trajectory of the arrow is noted.
Which graph best represents the vertical velocity
of the arrow as a function of time? Ignore air
resistance the only force acting is gravity.
Slide 2-39
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Answer
An arrow is launched vertically upward. It moves
straight up to a maximum height, then falls to
the ground. The trajectory of the arrow is noted.
Which graph best represents the vertical velocity
of the arrow as a function of time? Ignore air
resistance the only force acting is gravity.
D.
Slide 2-40
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Example You throw a ball into the air with speed
15.0 m/s how high does the ball rise?
Given viy 15.0 m/s ay ?9.8 m/s2
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• Also an how far question
• (vf2-vi2) 2ad
• What is vf? vi? What is a? (be careful of signs!)

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• You toss a ball straight up with an initial
  vi25m/s. You then become distracted. How long
  until the ball clunks you on your head?

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Example A penny is dropped from the observation
deck of the Empire State Building 369 m above the
ground. With what velocity does it strike the
ground? Ignore air resistance. How long will it
take to hit?
Given viy 0 m/s, ay ? 9.8 m/s2, ?y ?369 m
ay
Unknown vyf
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