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The magnitude of the momentum before and after the collsion is the same (m_bv_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.

=diag1} {color:red} *Diagrammatic Representation* {color}

{cloak:id=diag1}

{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||

{cloak:diag1}

{toggle-cloak:id=math1} {color:red} *Mathematical Representation* {color}

{cloak:id=math1}

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. 

{cloak:math1}
{card:Using Algebra}
{card:label=Adding Vectors to get Final Momentum}
h4. Adding to get the Final Momentum

We have defined impulse as the final [momentum|momentum] minus the initial [momentum|momentum], but subtracting [vectors|vector] can be confusing.  Therefore, we will first consider a rearrangement of the definition of impulse.  We can write:
\\
{latex}

\begin{large}\[ \vec{p}_{f} = \vec{p}_{i} + \vec{J} \] \end{large}{latex}
\\

!addimpulsevec.png!
{card:Adding Vectors to get Final Momentum}
{card:label=Subtracting Initial Momentum Vector from Final}

h4. Subtracting Initial Momentum from Final

It is also possible to draw a vector representation of the regular definition of impulse
\\
{latex}\begin{large}\[ \vec{J} = \vec{p}_{f} - \vec{p}_{i}  \] \end{large}{latex}
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but drawing a vector equation that includes subtraction is tricky.  We _must_ think of this equation in the following 

way:
\\
{latex}\begin{large}\[ \vec{J} = \vec{p}_{f} + (- \vec{p}_{i})  \] \end{large}{latex}
\\
In other