VIEWS: 8 PAGES: 19 POSTED ON: 4/7/2010
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 mv 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