Kinematics

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Kinematics
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Relative Motion

• Motion is relative
• The same event, viewed from two different points
  of view, can yield two different measurements

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Quantities of Interest

• Position where you are relative to a specific
  origin
• Elapsed Time measurement of a clock
• Speed/Velocity
• Acceleration

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Distance/Displacement

• Distance your entire trip
• Displacement difference between initial and
  final positions
• If you backtrack, or travel in multiple
  directions, these two numbers will be different

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Speed/Velocity

• Speed depends upon distanced traveled
• Speed distance traveled/time
• Velocity depends upon displacement
• Velocity displacement/time
• Which has direction?

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• Assuming humans originated in Africa and migrated
  to other parts of the world, some time would be
  required for this to occur. At the modest rate
  of 1km/year, how many centuries would it take
  humans to migrate from Africa to China, some
  10,000km away?

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• Is it possible for an object to change velocities
  while holding a constant speed?

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• I travel 20 miles N in 30 minutes, then 60 miles
  south in 90 minutes. What is my average
  velocity?
• .33 miles/min or 20 miles/hour
• South

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• On a car trip, I travel at 60 miles/hour for 2
  hours, stop and rest for 30 minutes, then travel
  at 70 miles/hour for 4 hours. How far do I
  travel? What is my average speed? (if youre
  feeling ambitious, draw a position/time graph for
  this trip)
• 400 miles
• 61.5 miles/hour

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Turnpike Tickets

• Regardless of how sneaky you might be, its
  possible to get caught speeding on turnpikes
  where you pick up a ticket at the entrance and
  drop off the ticket at the exit
• How?

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Average vs. Instantaneous

• Average quantities corresponding to lengths of
  time
• Instantaneous quantities correspond to instants
  in time
• Mathematically, were looking at the limit of a
  function as Dt approaches 0

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Instantaneous Velocity

• An objects velocity at a particular instant in
  time
• We can figure out its instantaneous velocity by
  looking at a position/time graph
• As we compute position and time differences over
  shorter and shorter intervals, we approach our
  instantaneous value (see Walker, p23)

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• As our time interval decreases, our slope
  (velocity) approaches a constant value
• Why?
• All functions, even curves, are linear on small
  enough scales

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Speed?

• The magnitude of our instantaneous velocity is
  instantaneous speed
• Were all familiar with a device that measures
  instantaneous speed
• What is it?

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Graphical Views of Motion

• Stationary (on x and v graphs)
• Constant speed (on both)
• Speeding up (on both)
• Slowing down (on both)
• Moving backwards (on both)

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Acceleration and Freefall
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Questions of Significance

• If you drop a penny off the Empire state
  building, how fast is moving when it hits the
  ground?
• If you launch an object into the air, how long
  does it take to hit the ground?
• How can you calculate a snow-boarders hang time?
  LeBron James hangtime?
• What is the minimum length necessary for an
  airport runway?

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Velocity/Time Graphs
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Slopes

• On a position/time graph, the slope represents
  the objects velocity
• How about the slope on a velocity/time graph?
• Slope rise/run
• Slope Dv/Dt

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Slope Units?

• Slope Dv/Dt
• Slope m/s/s m/s2
• What does the slope physically represent?
• The rate of change in velocity
• We call this quantity, acceleration
• Like velocity, it is a vector quantity

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The Meaning of Acceleration Units

• m/s/s what does this mean?
• Lets think about gravity, which accelerates
  objects at about 10m/s/s
• If you drop an object from rest, how fast will it
  fall?
• After the first second, 10m/s the second 20 m/s
  the 3rd, 30m/s

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Acceleration Values

• Acceleration due to gravity 9.8 m/s/s
• Honda Civic 3.0 m/s/s
• Jumbo Jet 2.5 m/s/s
• Space Shuttle 20 m/s/s

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Negative Acceleration

• Acceleration is a vector, which means it has a
  direction
• If I travel in the direction, but my
  acceleration is in the negative direction, what
  happens?
• Ex braking, throwing keys in the air

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Displacement?

• We know the relationship between acceleration and
  velocity
• How does displacement fit into the picture?
• Ex How far does a car travel as it accelerates
  from 0 to 60mph?

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• Imagine a rock, thrown downwards off a cliff at a
  speed of 30m/s
• I start my clock when the rock is 2m below the
  edge of the cliff
• Fill in the following table of information
  concerning this rock

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Time (s) Inst. Velocity (m/s) Average Velocity (s) Dx (m) Position (m)
0
1
2
3
4
5
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Time (s) Inst. Velocity (m/s) Average Velocity (m/s) Dx (m) Position (m)
0 30 ____ ___ 2
1 40 35 35 37
2 50 45 45 82
3 60 55 55 137
4 70 65 65 202
5 80 75 75 277
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• Graph position data vs. time data for this fall
  period and fit with the appropriate function
• How does position depend upon time?

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• Equation of the fit
• y(t) 5t2 30t 2
• What do the fit coefficients physically
  represent?

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• y final position
• 2 initial position (meters)
• 30 initial velocity (m/s)
• 5 ½ acceleration (m/s2)

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• yf 1/2aDt2 viDt yi
• yf yi 1/2aDt2 viDt
• Dy viDt 1/2aDt2

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Applying DVATs

• How far does a Porsch travel if it accelerates
  from 0 30 m/s (60 mph) over a time interval of
  6s?

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The Deadly Penny

• Will a penny, dropped from the Empire State
  building, kill someone on the ground below?

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How to Solve Physics Problems

• 1. Draw a picture (with initial and final)
• 2. Think about the following questions
• What am I trying to find?
• What do I know?
• What do I need to know?
• 3. Think about the physics at play
• 4. Find the appropriate mathematics
• 5. Solve it (in symbols first!)
• 6. Does your answer make sense?

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• What are we trying to find?
• Final velocity of a penny
• What do we know?
• Initial velocity, acceleration due to gravity
• What do we need to know?
• Height of building, fatal drop speed

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Useful information

• Empire State Building Height
• Height 381 m
• Velocity of a bullet
• Velocity 300 400 m/s

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DVAT Equations

• 1 vf vi aDt
• 2 Dx viDt ½ aDt2
• 3 Dx ½(vfvi) Dt
• 4 2aDx vf2-vi2

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The Catch 22

• The underlying assumption of these equations is
  constant acceleration
• If we dont have constant acceleration, we cant
  use them

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Situation 2 Human Acceleration

• Asafa Powell, the worlds fastest human,
  accelerates at a rate of 5m/s/s over a distance
  of 10m
• Assuming he starts from rest, what is his final
  velocity?

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Situation 3 Braking Distance

• According to the Highway Patrol, it takes about
  75m to slow down from 70mph (35m/s) on dry road
  conditions
• What is your braking acceleration?
• How long does it take to stop?
• If you are traveling at 20m/s, how much braking
  distance do you need?

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Impact Speed

• Lets revisit the previous situation
• Say you only have 20m before you hit the car in
  front of you (initial velocity 35m/s)
• At what speed will you hit the car?
• How does human reaction time change these
  numbers?

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Two Cars in Motion?

• Two cars, separated by 30m, both slam on their
  brakes at the same time
• Car 1, initially traveling at 35m/s, has an
  acceleration of -4.0m/s/s
• Car 2, initially traveling at 20m/s, has an
  acceleration of -8.0m/s/s
• At what speed will car 1 strike car 2?

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Free Fall

• Neil Armstrong video on youtube
• http//www.youtube.com/watch?v5C5_dOEyAfk

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Problem Types

• How high? (max height on a toss)
• How long? (drop/hang time)
• How fast? (final velocity)

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• Lebron James takes off from the ground with a
  vertical velocity of 5.5m/s
• How long is he in the air?
• How high does he go?

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Assumptions

• vf 0 at apex
• vi -vf (assuming starting and ending heights
  are the same)
• Acceleration -10m/s/s
• Time up Time down

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• In a fit of rage, a student leans out his 3 story
  window (10m in height) and throws his physics
  textbook straight up with an initial velocity of
  8m/s. Assuming this students rage has sucked in
  all the surrounding air
• How long will it take for the book to hit the
  ground below?
• At what speed will it strike the ground?

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A person standing by the edge of a cliff throws
one ball straight up and another straight down at
the same initial speed. Neglecting air, the ball
to hit the ground with the greater speed is the
one initially thrown

• 1. upward
• 2. downward
• 3. neitherthey both hit the ground at the same
  speed

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The graph on the following page maps the position
of two trains, A and B. Which statement below is
true?
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• 1. At time tB, both trains have the same
  velocity
• 2. Both trains have the same speed at all times
• 3. Both trains have the same velocity at some
  time before tB
• 4. Somewhere on the graph, both trains have the
  same acceleration

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The position/time graph on the next page maps the
motion of 4 objects. Answer the following
questions related to these objects motion
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Rank the objects average velocities in
increasing order

• 1. A, B, C, D 2. B, A, D, C
• 3. A, C, D, B 4. B, D, C, A
• 5. C, A, D, B

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Which object has the highest instantaneous
velocity (at any point during the time interval?)

• 1. A 2. B
• 3. C 4. D

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Two identical objects are dropped from different
heights. Object 1, dropped from height h,
reaches a speed v when it hits the ground.
Assuming object 2 is dropped from height 2h, how
fast is it traveling when it hits the ground?

• 1. v/2 2. v2v
• 3. 2v 4. 4v

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If I throw an object up in the air at speed v, it
rises 6m above my hand. If I throw that same
object on the moon (a 1.6 m/s/s) with speed v,
how high will it travel?
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A speeding car traveling at a constant 30m/s
passes a cop, initially at rest. If the cop
accelerates uniformly at 4m/s/s, how long does
it take him to catch the speeder?
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A stone is dropped from the roof of a tall
building. 1.0s later, a second stone is dropped.
How far apart are the stones when the second one
has reached a speed of 15.0m/s
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