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{excerpt:hidden=true}Simple problem illustrating the definition of [impulse] and the utility of an [initial-state final-state diagram].

Consider 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).  What is the impulse delivered to the ball in the collision?

h4. Solution

{toggle-cloak:id=sys} *System:* {cloak:id=sys} The ball as a [point particle].{cloak:sys}

{toggle-cloak:id=int} *Interactions:* {cloak:id=int} During the impact, we assume that the [collision force] from the wall is vastly larger than any other [external forces|external force] on the ball, so that other forces are ignored.{cloak:int}

{toggle-cloak:id=mod} *Model:* {cloak:id=mod}[Momentum and External Force].

{toggle-cloak:id=app} *Approach:*

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{toggle-cloak:id=diag} {color:red} *Diagrammatic Representation* {color}

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{warning}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 [initial-state final-state diagram] below (taking special note of the coordinate system) shows the resolution to this difficulty.{warning}

|!ballreversei.png!|!ballreversef.png!|
||Initial State||Final State||

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{toggle-cloak:id=math} {color:red} *Mathematical Representation* {color}

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The ball's initial _x_ momentum is positive in our coordinates (+{_}m{~}b{~}v{~}b{~}{_}), while its final _x_ momentum is _negative_ ( -- {_}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. 

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