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Impacts and There Graphs

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					Impacts and Their Graphs

   A closer look at Impulse
       Horizontal Impulse in Running
                          +    Net Positive Accel. During Contact
            0                  Positive impulse > negative impulse

Force (N)
                               No Net Acceleration During Contact
            0                  Positive impulse = negative impulse



                               Net Negative Accel. During Contact
            0   _              Positive impulse < negative impulse

                                Impulse is the area under the
                    Time (s)    force-time curve.
              Understanding Impulse
                                           F = ma
                                           +force       +acceleration

Horizontal
Force (N)                       Time (s)

                         F = ma
                         -force        - acceleration
    At each instant in time during a contact, a force acts to produce an
    acceleration. The Impulse is the net effect of all those
    instantaneous forces. In other words, it is the average force
    multiplied by the total time over which the forces have acted.

    Impulse = Average Force x Time force was applied or Fdt
    Impulse produces Mass x Change in Velocity or      mv
           Running Contact
• During a single running contact, your body
  undergoes both positive and negative forces
  that produce positive and negative
  accelerations.
• A force acting for a period of time produces
  an impulse.
• If the positive and negative impulses cancel
  each other out (equal areas), then the net
  impulse is zero and the runner is moving at
  a constant speed.
              Equal and Opposite
  When a ball hits a bat, or a foot lands on the ground, there
  are equal and opposite impulses applied to both objects. The
  force-time curve of one is the inverted force time curve of
  the other.



                             Ball
Force (N)
                                         Time (s)
                              Bat
                                     Peak forces are equal and
                                     occur at the same time
             Vertical Impulse: Countermovement jump
                                                             mg

                                                             Fy

                  Fy
External Forces




                                                      Time
                   -mg


                             Fy = Fy – mg = may
                     Combining Fy and -mg gives the graph of the sum of the
                     forces in the vertical direction
Resultant Forces (Fy)




                                             a
                                                                       e
                                start
                                                                       Time
                                                       b
                           a:   Bottom of Squat            c   d
                           b:   Take-Off
                           c:   Peak Jump Height
                           d:   Landing
                           e:   Back to Standing
Acceleration of Center of Mass




                                                                                Time




                                 Where is the greatest negative velocity?
                                 Where is the greatest positive velocity?
                                 At what point is the jumper’s velocity zero?
                            tf
            Vf = Vi + aydt
                           ti

• The equation states that final velocity is
  equal to initial velocity plus the change in
  velocity.
• Where the change in velocity is the net area
  under the acceleration curve.
  Relative Velocity of Impacting
             Bodies
• When dealing with impact, we need to
  know what the velocities of the objects are
  relative to each other.
• To determine relative velocity, you subtract
  one velocity from the other.
• Essentially, you are saying if one object’s
  velocity was zero, what would the velocity
  of the other object be?
• V–v
Relative Velocity: Heading a Soccer Ball

                     -10 m/s

                +4 m/s


                               Ball relative to Player

                               -10 - (+4) = -14 m/s

                               Player relative to Ball

                               +4 - (-10) = +14 m/s
   Coefficient of Restitution (e)
• It is a measure of the elasticity of two
  objects that impact. Values can range from
  0 (no bounce) to almost 1 (highest bounce).
• It is calculated using the ratio of the relative
  velocity of separation to the relative
  velocity of approach.
• e = -(V–v)                  ..… 1
       (V–v)
 It can also be measured as the ratio of the restitution
 impulse to the compression impulse.
Conservation of Linear Momentum
• The forces of impact are internal to the system
  (M + m). The above law states that if no external
  forces act on the system, then the momentum of the
  system is the same before and after impact.
• MV + mv = MV + mv            ….. 2
• Combining equations 1 and 2, we can solve for the
  velocities of the objects post impact.

• v = M[ V(1+e) – ev] + mv           ….. 3
             M +m
                   Special Case
• In certain cases, when one of the objects is
   at rest prior to impact (such as golf), the
   equation can be simplified. The velocity of
   the golf ball is:
 • v = MV(1+e)
           M+m

Where: v is the velocity of the golf ball after impact, m is the
mass of the golf ball, V is the velocity of the clubhead prior to
impact, M is the mass of the clubhead, and e is the COR.
        Applying the Formula
• Would a golf ball go farther if you
  increased clubhead mass, or increased it’s
  velocity?
• We can graphically display how each of
  these variables would affect the velocity of
  the golf ball after impact.
                Increasing Clubhead Mass
                    Max Ball Velocity
Ball Velocity




                    Clubhead Mass

M is on both the top and bottom of the equation, so when M gets
large, increases in M show very small changes in ball velocity
Increasing the Velocity of the Club

                                       Ball velocity increases as
                                       club velocity increases.
Ball Velocity




                  Velocity of Golf Club

V is only on the top of the equation, so the bigger it gets, the
bigger ball velocity gets.
Clubhead Mass Clubhead Velocity Ball Mass e     Ball Velocity
          0.5               40        0.05 0.83         66.5
            1               40        0.05 0.83         69.7
            2               40        0.05 0.83         71.4
           10               40        0.05 0.83         72.8
         1000               40        0.05 0.83         73.2
          0.5               40        0.05 0.83         66.5
          0.5               45        0.05 0.83         74.9
          0.5               50        0.05 0.83         83.2
          0.5               70        0.05 0.83        116.5
          0.5             1000        0.05 0.83       1663.6
 What is the Clubhead Velocity?
With what velocity must a 0.25 kg clubhead contact a
0.05 kg golf ball in order to send the ball off the tee
with a velocity of 80 m/s? e = .81

  v = MV(1+e)          Therefore,
       M+m

				
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posted:4/7/2010
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