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h1. Impulse

{excerpt}The time integral of [force|force].  The net [external|external force] impulse acting on a [system|system] over a given time interval is equal to the system's change in [momentum|momentum].{excerpt}

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h3. {toggle-cloak:id=mot} Motivation for Concept

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Although some everyday [interactions|interaction] like [gravity|gravitation] and [friction|friction] result in stable [forces|force] whose effects can easily be analyzed with [dynamics|Point Particle Dynamics], many interactions are not steady.  Consider, for example, the difference between a push and a punch.  When you push something, you consciously use your muscles to apply a steady [force|force] to the target object.  For this reason, pushing a bowling ball or a bean bag feels much the same, apart from the fact that the bean bag will probably deform more in response to the [force].  When you punch something, however, you simply let your fist fly.  The [force|force] felt by the target object and the [reaction force|Newton's Third Law] exterted on your fist are the result of the impact of your moving fist with the object.  This impact is essentially out of your control.  The [force] exterted are not determined directly by your muscles (though the faster your fist is moving, the greater the force will tend to be), but rather it is principally determined by the properties of your hand and the target object.  Because of this fact, there is a dramatic difference between the results of punching a bean bag versus punching a bowling ball.

Such impact or collision forces are extremely common in everyday life.  Almost any sport will involve collisions.  Household activities like hammering nails or kneading dough require collisions.  Understanding collisions is also of great importance to car manufacturers.  

Unfortunately, the forces during a collision are very difficult to characterize.  They change extremely rapidly in time (the entire duration of a typical collision is measured in milliseconds), and they manner in which they change is stronly dependent on the material properties of the objects undergoing the collision.  Because of these complications, it is rare to see a detailed force profile for a collision.  Instead, collisions are usually described by an effect that is much more easily observed:  the resulting change in the motion of the participants.  The impulse delivered by a collision is one measure of this change.  

|!ball03.jpg!!ball04.jpg!!ball05.jpg!!ball06.jpg!!ball07.jpg!!ball08.jpg!|
|Photos taken with a high-speed digital camera showing the impact of a (new) tennis ball dropped from a height of 100 inches onto a wooden platform.\\The images shown are spaced by approximately 1 ms.\\Special thanks to James Bales at the [MIT Edgerton Center|http://web.mit.edu.ezproxyberklee.flo.org/edgerton] for helping Andrew Pawl to capture the image.\\For footage of tennis ball impacts at _much_ higher speeds see the [ITF website|http://www.itftennis.com/technical/].|
{note}Note that the center images are less blurry than the first and last.  Can you explain this?{note}
{info}The ball rebounds because it is elastic.  The deformation of the ball observed in the center image is not permanent.  The ball's structure causes it to return to its initial shape, and in the process it pushes itself off the ground.  If the ball were made of clay, the deformation would remain and the ball would simply "splat" onto the ground (one more example of how structural differences complicate collision analysis).{info}

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h3. {toggle-cloak:id=def1} One Definition of Impulse

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h4. {toggle-cloak:id=vecchange} Vector Change in Momentum

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Suppose an object experiences a sudden [interaction|interaction] that results in a dramatic change in the object's [momentum|momentum].  One definition of the impulse provided by the [interaction|interaction] is to calculate the numerical value of the change in the object's [momentum|momentum].  In other words, the impulse _J_ is:
{latex}\begin{large} \[ \vec{J} = \vec{p}_{f} - \vec{p}_{i} = m\vec{v}_{f} - m\vec{v}_{i}\]\end{large}{latex}

Note that because momentum is a vector, the change in [momentum|momentum] is also a [vector|vector].  Thus, the impulse is by defintion a [vector|vector] quantity.  As with any [vector|vector] quantity, it is important to remember that the calculation of impulse really involves three equations, one for each component:
{latex}\begin{large}\[ J_{x} = mv_{f,x} - mv_{i,x}\]
\[ J_{y} = mv_{f,y} - mv_{i,y} \]
\[ J_{z} = mv_{f,z} - mv_{i,z} \]\end{large}{latex}

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h4. {toggle-cloak:id=initfin} Initial-State Final-State Diagrams

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The vector nature of [momentum|momentum] means that it very important when calculating the change to carefully set up a coordinate system.  For this reason, it is strongly recommended that you begin any problem involving a change in [momentum|momentum] by constructing an [intial-state final-state diagram].  

Consider, for example, a ball of mass _m{~}b{~}{_} that is moving to the right at a constant speed _v{~}b{~}{_} when it suddenly impacts a wall and reverses direction (still moving at the same speed).  The [{_}magnitude{_}|magnitude] of the [momentum|momentum] before and after the collsion is the same ({_}m{~}b{~}v{~}b{~}{_}), which can easily lead to the conclusion that there has been no change.  Thinking about the situation, however, should quickly convince you that the ball has certainly been acted on by some force, which implies that a change _did_ occur.  Carefully drawing the diagram below shows the resolution to this difficulty.

The ball's initial _x_ momentum was +{_}m{~}b{~}v{~}b{~}{_}, while its final _x_ momentum is -{_}m{~}b{~}v{~}b{~}{_}, giving a change of:
{latex}\begin{large}\[ J_{x} = -m_{b}v_{b} - m_{b}v_{b} = 2m_{b}v_{b}\]\end{large}{latex}
where the negative sign indicates that the impulse is applied in the negative _x_ direction, and so the impulse points leftward in this case. 
{note}If the leftward direction for the impulse does not make sense to you, read through the next section (Integral Form of Newton's Second Law).{note}

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h3. Integral formForm of Newton's Second Law

h3. {toggle-cloak:id=timeav} Impulse and Time-Averaged Force

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{warning}It is also common in mechanics to discuss a [{_}distance{_}|distance]-averaged [force|force] found using the definition of [work|work].  These different averages will _not_ necessarily yield the same value!  (Though they will, of course, give the same answer for the special case of a constant force.){warning}

{info}To see why different averages need not give the same value, consider two babysitters who each make $15 per hour.  They have the same _time-averaged_ income.  Suppose, however, that one of the sitters gets jobs that last 3 hours on average, while the other's jobs are 4 hours.  Their _job-averaged_ income is different ($45 vs. $60).  Similarly, you could imagine two people who mow lawns for extra income, charging $40 per lawn.  Their job-averaged income is the same, but their time-averaged income could be different if they tend to mow different sized lawns.{info}
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