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|!Boxcar and Cannonballs 01.PNG!|


Imagine that you have an indestructible boxcar sitting on frictionless railroad track. The boxcar has length *L*, height *H*, and width *W*. It has *N* cannonballs of radius *R* and mass *M* stacked up against one end. If I move the cannonballs in any fashion -- slowly carrying them, rolling them, firing them out of a cannon -- what is the furthest I can move the boxcar along the rails? Which method should I use to move the boxcar the furthest? Assume that the inside walls are perfectly absorbing, so that collisions are perfectly inelastic.

h2. Part A


h4. Solution One

{toggle-cloak:id=sysa} *System:*  {cloak:id=sysa}Boxcar and cannonballs as [point particles|point particle].{cloak}

{toggle-cloak:id=inta} *Interactions:* {cloak:id=inta} Not Important in this part.{cloak}

{toggle-cloak:id=moda} *Model:*  {cloak:id=moda}[Point Particle Dynamics].{cloak}

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

{cloak:id=appa}

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

{cloak:id=diaga}


|!Boxcar and Cannonballs 02.PNG!|
|!Boxcar and Cannonballs 03.PNG!|

The system consists of the Boxcar on rails and the Cannonballs, plus whatever devices we use for propulsion inside.There are thus no external influences

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{toggle-cloak:id=matha} {color:red} *Mathematical Representation* {color}
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Since there are no external influences, which includes forces, the [Center of Mass] of the system is not affected, and by the Law of Conservation of [Momentum] must remain fixed. . 

{latex}\begin{large}\[\ M_{Boxcar} x_{Boxcar, initial} + \sum M_{i} x_{i, initial} = M_{Boxcar} x_{Boxcar, final} + \sum M_{i} x_{i, final}  \]\end{large}{latex}

Here {*}x{~}i{~}{*}  is the position of the center of the *i*th cannonball and {*}x{~}Boxcar{~}{*} is the position of the center of the boxcar. The subscripts *initial* and *final* indicate the positions at the start and the end of our operation.  

Re-arranging, we get that the final boxcar position is:

{latex}\begin{large}\[ x_{Boxcar, final} = x_{Boxcar, initial} + \frac{\sum{M_{i} x_{i, initial}} - \sum{M_{i} x_{i, final}}}{M_{Boxcar}} \]\end{large}{latex}
\\
{latex}\begin{large}\[ x_{Boxcar, final} = x_{Boxcar, initial} - \frac{M_{i}}{M_{Boxcar}} ( \sum{x_{i, final}} - \sum{x_{i, initial}}) \]\end{large}{latex}
\\
The location of the Center of Mass *<x>* is given by:

{latex}\begin{large}\[ < x > = \frac{M_{Boxcar} x_{Boxcar,initial} + \sum{M_{i}x_{i,initial}}}{M_{Boxcar} + \sum{M_{i}}} \]\end{large}{latex}


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h2. Part B




Now consider the intermediate stages as the cannonballs are moved slowly, one by one, from one side to the other. How does the car move as each ball is shifted?
h4. Solution

{toggle-cloak:id=sysb} *System:* {cloak:id=sysb}Boxcar and cannonballs as [point particles|point particle].{cloak}

{toggle-cloak:id=intb} *Interactions:* {cloak:id=intb}None.{cloak}

{toggle-cloak:id=modb} *Model:*  {cloak:id=modb}[Point Particle Dynamics].{cloak}

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

{cloak:id=appb}

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

{cloak:id=diagb}

 
|!Boxcar and Cannonballs 02.PNG!|
|!Boxcar and Cannonballs 04.PNG!|
|!Boxcar and Cannonballs 05.PNG!|
|!Boxcar and Cannonballs 06.PNG!|
|!Boxcar and Cannonballs 07.PNG!|
|!Boxcar and Cannonballs 08.PNG!|
|!Boxcar and Cannonballs 03.PNG!|


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

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We calculate the Center of Mass as each Cannonball is shifted from one side to the other. Assume that each ball moves from as close to one side of the boxcar to as far one the other side as it can go.



{latex}\begin{large}\[\sum F_{x} = F_{A}\cos\theta - N = ma_{x}\]
\[ \sum F_{y} = F_{A}\sin\theta - mg = ma_{y}\]\end{large}{latex}



{latex}\begin{large}\[ N = F_{A}\cos\theta = \mbox{150 N} \]\end{large}{latex}

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h2. Part C

!thatnormal3.jpg|width=500!

Now imagine that the cannonballs are fired from one end to the other, one by one. What are the equations of motion during and after, and how does the car move along?

h4. Solution

{toggle-cloak:id=sysc} *System:*  {cloak:id=sysc}Boxcar and cannonballs as [point particles|point particle].{cloak}

{toggle-cloak:id=intc} *Interactions:* {cloak:id=intc}Conservation of Momentum and equal and opposite forces.{cloak}

{toggle-cloak:id=modc} *Model:*  {cloak:id=modc}[Point Particle Dynamics].{cloak}

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

{cloak:id=appc}

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

{cloak:id=diagc}



{cloak:diagc}

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

{cloak:id=mathc}

We write the equations of motion for the boxcar (with the rest of the cannonballs) and the one fired cannonball.

{latex}\begin{large}\[\sum F_{x} = F_{A}\cos\theta = ma_{x}\]
\[ \sum F_{y} = F_{A}\sin\theta - mg - N = ma_{y}\]\end{large}{latex}



{latex}\begin{large}\[ F_{A}\sin\theta  - mg - N = 0 \]\end{large}{latex}



{latex}\begin{large}\[ N = F_{A}\sin\theta - mg = \mbox{52 N}\]\end{large}{latex}



{cloak:mathc}
{cloak:appc}