Consider a ball of mass mb that is moving to the right at a constant speed vb 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?
Solution
System: The ball as a point particle. Interactions: During the impact, we assume that the collision force from the wall is vastly larger than any other external forces on the ball, so that other forces are ignored. Model: Momentum and External Force. Approach:We will solve this problem using three different approaches to illustrate alternate ways to perform vector subtraction.
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{excerpt:hidden=true}Simple problem illustrating the definition of [impulse] and the utility of an [initial-state final-state diagram].{excerpt}
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].{cloak:mod}
{toggle-cloak:id=app} *Approach:*
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We will solve this problem using three different approaches to illustrate alternate ways to perform vector subtraction.
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{card:label=Using Algebra}
h4. Using Algebra
{toggle-cloak:id=diag1} {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}
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{toggle-cloak:id=math1} {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:
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The magnitude of the momentum before and after the collsion is the same (mbvb), 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. |
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The ball's initial x momentum is positive in our coordinates (+mbvb), while its final x momentum is negative (-- mbvb), giving a change of:
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\begin{large}\[ J_{x} = -m_{b}v_{b} - m_{b}v_{b} = -2m_{b}v_{b}\]\end{large} |
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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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{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:
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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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Adding to get the Final MomentumWe have defined impulse as the final momentum minus the initial momentum, but subtracting vectors can be confusing. Therefore, we will first consider a rearrangement of the definition of impulse. We can write:
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\begin{large}\[ \vec{p}_{f} = \vec{p}_{i} + \vec{J} \] \end{large} |
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[vector|] __ [|momentum] [|momentum].
We can use this formulation to draw a vector diagram representing the ball-wall collision. Remembering the rules for adding [vectors|vector] . We can use this formulation to draw a vector diagram representing the ball-wall collision. Remembering the rules for adding vectors tail-to-tip, |
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{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
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| Adding Vectors to get Final Momentum |
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| label | Subtracting Initial Momentum Vector from Final |
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Subtracting Initial Momentum from FinalIt is also possible to draw a vector representation of the regular definition of impulse
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\begin{large}\[ \vec{J} = \vec{p}_{f} - \vec{p}_{i} \] \end{large} |
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but drawing a vector equation that includes subtraction is tricky. We _must_ think of this equation in the following way:
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but drawing a vector equation that includes subtraction is tricky. We must think of this equation in the following way:
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\begin{large}\[ \vec{J} = \vec{p}_{f} + (- \vec{p}_{i}) \] \end{large} |
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In other __ __
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which gives the same impulse vector as the diagram above.
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which gives the same impulse vector as the diagram above.
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