Kinematics

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Kinematics Linear Motion

• Kinematics
• and
• Linear Motion
• Updated 9/10/07

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Kinematics Linear Motion

• kine From the Greek kinein, to move
• AND
• matics relating to
• GIVING
• kinematics relating to motion

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Kinematics Linear Motion

• The kinematics topics we will study include
• distance and displacement (d or s)
• speed and velocity (u or v)
• constant acceleration (a)
• free fall due to gravity (g)

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Kinematics Linear Motion

• distance how far an object moves.
• drive a mile in a straight line, the distance
  traveled is one mile.
• drive a mile around a circular track, the
  distance traveled is one mile.
• units are meters, kilometers or miles

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Kinematics Linear Motion

• Example calculating distance
• Starting at CBHS, we drive 2 km east to A1A, then
  10 km south to the Pineda Causeway, then 6 km
  west to US1, then 17 km north to the 520, then 7
  km south to Minuteman Causeway, then 2 km west to
  CBHS. What is the total distance we have driven?

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Kinematics Linear Motion

• How fast we move a distance is speed.
• In other words, speed is how far an object
  travels in a certain amount of time.
• drive 50 miles in a straight line in one hour,
  the speed will be 50 mph.
• drive 50 miles around a circular track, the speed
  will be 50 mph.

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Kinematics Linear Motion

• speed (v) distance (meters)
• ------------------------
  --------------------------------------
• time (seconds)
• In symbols
• v d (m)
• -------
  --------------
• t (s)

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Kinematics Linear Motion

• Example calculating speed
• Starting at CBHS, we drive 2 km east to A1A, then
  10 km south to the Pineda Causeway, then 6 km
  west to US1, then 17 km north to the 520, then 7
  km south to Minuteman Causeway, then 2 km west to
  CBHS. If it takes 1 hour, what is the average
  speed?

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Kinematics Linear Motion

• displacement how far an object is from its
  point of origin.
• drive a mile in a straight line, the displacement
  is one mile.
• drive a mile around a circular track, the
  displacement is zero miles.
• units are meters, kilometers or miles

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Kinematics Linear Motion

• For movement in a straight line distance and
  displacement are the same thing.
• Using xi to mark the starting position and xf
  the finishing position
• displacement ?x xf - xi

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One-Dimensional Coordinates
We usually use x to indicate position, then xi
would be the initial position and xf would be the
final position. The displacement traveled (?x)
is d ?x xf - xi
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The Kingfisher Takes a Plunge
For vertical displacement, we may use y instead
of x d ?y yf - yi
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Kinematics Linear Motion

• Example calculating displacement
• Starting at CBHS, we drive 2 km east to A1A, then
  10 km south to the Pineda Causeway, then 6 km
  west to US1, then 17 km north to the 520, then 7
  km south to Minuteman Causeway, then 2 km west to
  CBHS. What is the total displacement?

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Kinematics Linear Motion

• velocity how far an object moves (the
  displacement) from its point of origin in a
  certain time.
• drive 50 miles in a straight line in and hour,
  the velocity will be 50 mph.
• drive 50 miles in one hour around a one mile
  circular track, and the velocity will be 0 mph.

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Kinematics Linear Motion

• Velocity using xi to mark the starting
  position, ti the starting time, xf the finishing
  position, and tf the finishing time
• velocity xf - xi ? x
• 
  ------------------------- ----------------
• tf - ti ? t

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Kinematics Linear Motion

• velocity displacement (meters)
• -------------------------------
  --------------------------------------------------
  -------------
• time (seconds)
• In symbols
• v d (m)
• -------
  --------------
• t (s)

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Kinematics Linear Motion

• Example calculating velocity
• Starting at CBHS, we drive 2 km east to A1A, then
  10 km south to the Pineda Causeway, then 6 km
  west to US1, then 17 km north to the 520, then 7
  km south to Minuteman Causeway, then 2 km west to
  CBHS. If it takes 1 hour, what is the average
  velocity?

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Kinematics Linear Motion

• Average Speed
• Besides v ? x / ? t if the acceleration is
  constant, another way to calculate the average
  speed is
• velocity vi vf
• 
  -------------------------
• 2

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Kinematics Linear Motion

• Kinematics Units
• Speed and velocity ( preferred)
• mph (miles per hour)
• km/h, kmh-1 (kilometers per hour)
• m/s, m s-1 (meters per second)

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Kinematics Linear Motion

• Graphing Motion (Preview)
• position vs time slope is speed area has no
  meaning.
• velocity vs time slope is acceleration area is
  displacement.
• acceleration vs time slope is rate of
  acceleration area is speed.

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Constant Velocity on an x-Versus-t Graph
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Kinematics Linear Motion

• So far, the speed, velocity and acceleration we
  have been discussing are averages over a time
  interval.
• Instantaneous speed and velocity are the same as
  the average speed and velocity if the speed and
  velocity do not change (are constant) over the
  time interval.

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Motion Along the X Axis Represented with an
x-Versus-t Graph
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Average Velocity on an x-Versus-t Graph
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Average Velocity on an x-Versus-t Graph
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Instantaneous Velocity
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Graphical Interpretation of Average and
Instantaneous Velocity
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Kinematics Linear Motion

• Instantaneous velocity
• v lim ? x (or dx/dt)
• ?t ? 0 --------------
• ? t
• Instantaneous acceleration
• a lim ? v (or dv/dt)
• ?t ? 0 --------------
• ? t

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Kinematics Linear Motion

• acceleration acceleration is about a change of
  speed or velocity. In other words, how fast are
  we speeding up, slowing down, or changing
  direction?
• if we go from 0 m/s to 20 m/s in 10s, the
  acceleration is 2 m/s2
• note that the units are m/s2 since we are
  dividing m/s by s.

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Kinematics Linear Motion

• Acceleration Using vi to mark the starting
  velocity, ti the starting time, vf the finishing
  velocity, and tf the finishing time
• acceleration vf - vi ? v
• 
  ------------------------- ----------------
• tf - ti ? t

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Kinematics Linear Motion

• acceleration velocity
• -----------------------
  --------------
• time
• In symbols
• a v (m/s)
• -------
  --------------------
• t (s)

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Kinematics Linear Motion

• Instantaneous acceleration is the same as the
  average acceleration if the acceleration does not
  change (is constant) over the time interval.

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Kinematics Linear Motion

• Kinematics Units
• Acceleration ( preferred)
• mph/s (miles per hour per second)
• km/h/s, km h-1s-1 (km per hour per sec)
• m/s2, m s-2 (meters per sec per sec)

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Kinematics Linear Motion

• distance and speed are scalars, that is they have
  magnitude (size), but not direction.
• velocity and acceleration are vectors, they imply
  both magnitude and direction.
• vectors are indicated in bold type, and sometimes
  have a line over the symbol (v), while scalars
  are shown in normal type (v).

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Kinematics Linear Motion

• Important Equations
• s vt (d vt)
• s ut ½at2 (d vit ½at2)
• v at v u at (v vi at)
• v2 u2 2as (v2 vi2 2ad)
• vavg (u v) / 2 (vavg (vivf)/2)

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Kinematics Linear Motion

• IB Equations
• Most of the important equations of motion can be
  found in the IB workbook on page 11.
• (Note The u is the same as vi.)

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Kinematics Linear Motion

• Gravity
• The acceleration due to gravity is normally
  represented by the symbol g. The value for g
  is 9.8 m/s2 (or sometimes rounded to 10 m/s2.)
• (Note Gravity varies slightly from place to
  place on earth, and may be quite different on
  different planets.)

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Values of gat Different Locations on Earth
(ms-2)
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Kinematics Linear Motion

• Gravity
• Gravity (g) can be used in any of the kinematics
  equations in place of the a for acceleration.

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Free Fall from Rest
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Bombs Away Calculating the Speed of a Lava Bomb
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Position and Velocity of a Lava Bomb
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Velocity and Acceleration of a Lava Bomb as
Functions of Time
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Kinematics Linear Motion

• Graphing Motion (Postview)
• position vs time slope is speed area has no
  meaning.
• velocity vs time slope is acceleration area is
  displacement.
• acceleration vs time slope is rate of
  acceleration area is speed.

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Kinematics Linear Motion

• End of
• Kinematics
• and
• Linear Motion