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A

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4460

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lb

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Ford

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Explorer

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traveling

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35

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mph

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has

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a

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head

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on

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collision

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with

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a

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2750

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lb

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Toyota

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Corolla,

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also

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traveling

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35

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mph.

Excerpt
hiddentrue

Compare the forces on the occupants of two cars in a 1-D totally inelastic collision.

Composition Setup
 Deck of Cards
idbigdeck
  

{excerpt:hidden=true}Compare the forces on the occupants of two cars in a 1-D totally inelastic collision.{excerpt}
{composition-setup}{composition-setup}


{deck:id=bigdeck}
{card:label=Part A}
h2. Part A

Assuming that the automobiles become locked together during the collision, what is the speed of the combined mass immediately after the collision?

h4. Solutions

{toggle-cloak:id=sysa} *System:*  {cloak:id=sysa}Explorer plus Corolla as [point particles|point particle].{cloak}

{toggle-cloak:id=inta} *Interactions:* {cloak:id=inta}[Impulse|impulse] resulting from [external influences|external force] will be neglected, as we assume that the collision is instantaneous.{cloak}

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

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

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

{cloak:id=diaga}
We begin by sketching the situation and defining a [coordinate system|coordinate system].

|!CollisionInit.jpg|width=325!|!CollisionFin.jpg|width=325!|
||Initial State||Final State||

{cloak:diaga}

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

{cloak:id=matha}

Since we assume that [external|external force] [forces|force] are negligible during the collision, we set the [external|external force] [impulse|impulse] to zero which gives:

{latex}
 Card
labelPart A
Part A
Assuming that the automobiles become locked together during the collision, what is the speed of the combined mass immediately after the collision?
Solutions
 Toggle Cloak
idsysa
 System: 
 Cloak
idsysa
Explorer plus Corolla as point particles.
 Toggle Cloak
idinta
 Interactions: 
 Cloak
idinta
Impulse resulting from external influences will be neglected, as we assume that the collision is instantaneous.
 Toggle Cloak
idmoda
 Model: 
 Cloak
idmoda
.
 Toggle Cloak
idappa
 Approach: 
 Cloak
idappa
 Toggle Cloak
iddiaga
  Diagrammatic Representation 
 Cloak
iddiaga
We begin by sketching the situation and defining a coordinate system.
Image Added
Image Added
Initial State
Final State
 Cloak
diaga
diaga
 Toggle Cloak
idmatha
  Mathematical Representation 
 Cloak
idmatha
Since we assume that external forces are negligible during the collision, we set the external impulse to zero which gives:
 Latex
\begin{large}\[ p^{TC}_{x,i} + p^{FE}_{x,i} = p^{system}_{x,f} \]\end{large}
{latex}


or, 
 
 in 
 
 terms 
 
 of 
 
 the 
 
 masses:


{
 Latex
}
\begin{large}\[ m^{TC}v^{TC}_{x,i} + m^{FE}v^{FE}_{x,i} = (m^{TC}+m^{FE})v_{x,f} \]\end{large}
{latex}


which 
 
 gives:


{
 Latex
}
\begin{large}\[ v_{x,f} = \frac{m^{TC}v^{TC}_{x,i} + m^{FE}v^{FE}_{x,i}}{m^{TC}+m^{FE}} = \mbox{3.71 m/s} = \mbox{8.3 mph}\]\end{large}
{latex}

{warning}Remember that in our [coordinate system|coordinate system], the Corolla has a negative x-velocity before the collision.{warning}

{cloak:matha}
{cloak:appa}
{card}
{card:label=Part B}

h2. Part B

Find the [impulse|impulse] that acted on each of the vehicles during the collision.

h4. Solution

{toggle-cloak:id=sysb} *Systems:* {cloak:id=sysb} Corolla and Explorer as *separate* [point particle] [systems|system].{cloak}

{toggle-cloak:id=intb} *Interactions:* {cloak:id=intb}The [impulse|impulse] on each vehicle from the other is assumed to be the dominant [interaction|interaction] during the collision.  Because we are now considering the vehicles separately, these are now [external|external force] [impulses|impulse].{cloak}

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

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

{cloak:id=appb}

With the results of Part A it is straightforward to calculate the [impulse|impulse] on the Explorer due to the collision.  To be specific, we label the impulse "EC" to remind ourselves the [impulse|impulse] on the Explorer is provided by the Corolla.

{latex}\begin{large}\[ J^{EC}_{x} = p^{E}_{x,f}-p^{E}_{x,i} = m^{E}(v^{E}_{x,f}-v^{E}_{x,i}) = (\mbox{2030 kg})(\mbox{3.71 m/s}-\mbox{15.6 m/s}) = -\mbox{24000 kg m/s} \] \end{large}{latex}

Similarly, for the Corolla:

{latex}\begin{large}\[ J^{CE}_{x} = p^{C}_{x,f}-p^{C}_{x,i} = m^{C}(v^{C}_{x,f}-v^{C}_{x,i}) = (\mbox{1250 kg})(\mbox{3.71 m/s}+\mbox{15.6 m/s}) = +\mbox{24000 kg m/s} \] \end{large}{latex}

{note}Again, it is important to note that the Corolla's initial x-velocity is negative in our chosen [coordinate system|coordinate system].{note}

{tip}It is no coincidence that _J{_}{^}EC^ = -- _J{_}{^}CE^.  The relationship is guaranteed by [Newton's 3rd Law|Newton's Third Law].{tip}

{cloak}
{card}
{card:label=Part C}

h2. Part C

Assuming the collision lasted for 0.060 seconds, find the [time-averaged|impulse] [force|force] exerted on each vehicle.

h4. Solution

{toggle-cloak:id=sysc} *Systems:* {cloak:id=sysc} Corolla and Explorer as *separate* [point particle] [systems|system].  {cloak}

{toggle-cloak:id=intc} *Interactions:* {cloak:id=intc} The [external|external force] [force|force] on each vehicle from the other is assumed to be the dominant [interaction|interaction] during the collision.{cloak}

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

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

{cloak:id=appc}
From the definition of [impulse|impulse] we know:

{latex}\begin{large}\[ J^{EC}_{x} = \int F^{EC}_{x}\:dt \equiv \bar{F}^{EC}_{x} \Delta t\]\end{large}{latex}

so the time-averaged [force|force] exerted on the Explorer is:

{latex}\begin{large}\[\bar{F}^{EC}_{x} = \frac{J^{CE}_{x}}{\Delta t} = -\mbox{400,000 N} \]\end{large}{latex}

Similarly, the time-averaged [force|force] exerted on the Corolla is:

{latex}\begin{large}\[\bar{F}^{CE}_{x} = \frac{J^{EC}_{x}}{\Delta t} = \mbox{400,000 N}\]\end{large}{latex}

{tip}Again, the relationship between the two [forces|force] is guaranteed by [Newton's 3rd Law|Newton's Third Law].{tip}

{cloak}
{card}
{card:label=Part D}


h2. Part D

Suppose a 75 kg person in each vehicle underwent the same change in [velocity|velocity] as their automobile in the same amount of time.  Find the [time-averaged|impulse] [force|force] exerted on these people.

{toggle-cloak:id=sysd} *Systems:* {cloak:id=sysd}First, the Corolla and Explorer as separate [point particle] systems, then the passengers as *separate* [point particle] systems.{cloak}

{toggle-cloak:id=intd} *Interactions:*  {cloak:id=intd} Each vehicle is subject to a collision [force|force] from the other.  The passengers are each subject to some decelerating [force|force], possibly a seatbelt or airbag.{cloak}

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

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

{cloak:id=appd}
The passengers clearly are not subject to the same [force|force] as their vehicles.  Rather, they are subject to the same [accelerations|acceleration].  (They are -- hopefully -- strapped into their automobiles, so that whatever happens to their vehicle's motion happens to their motion as well.)  Thus, our first goal is to determine the vehicles' [accelerations|acceleration].  This is easily accomplished using the results of Part C in [Newton's 2nd Law|Newton's Second Law].

{latex}\begin{large}\[ a^{E}_{x}= \frac{F^{EC}_{x}}{m^{E}} = -\mbox{200 m/s}^{2}\]
\[ a^{C}_{x} = \frac{F^{CE}_{x}}{m^{C}} = \mbox{320 m/s}^{2} \]\end{large}{latex}

{tip}The Corolla's [acceleration|acceleration] should clearly be larger, even though the beginning and ending speeds of the Explorer and Corolla are the same.  The reason is that the Corolla has _changed direction_.  Thus, it effectively dropped all the way to zero from 35 mph and then _accelerated_ back up to 8.3 mph the other way.  The Explorer simply dropped its forward speed from 35 mph to 8.3 mph.{tip}

{info}Note (discounting safety measures like airbags) the SUV driver's acceleration would be about 20 [g's|gee], near the limit of human endurance.  The Corolla's driver would experience almost 33 [g's|gee].{info}

With these [accelerations|acceleration], we can find the [force|force] on each driver:

{latex}\begin{large}\[ F^{E,driver}_{x} = m^{driver}a^{E}_{x} = -\mbox{15000 N} = -\mbox{3,300 lbs}\]
\[ F^{C,driver}_{x} = m^{driver}a^{C}_{x} = \mbox{24000 N} = \mbox{5,400 lbs}\]\end{large}{latex}

{cloak}
{card}
{deck}


{td}
{tr}
{table}
{live-template:RELATE license}





 Warning
Remember that in our coordinate system, the Corolla has a negative x-velocity before the collision.
 Cloak
matha
matha
 Cloak
appa
appa
 Card
labelPart B
Part B
Find the impulse that acted on each of the vehicles during the collision.
Solution
 Toggle Cloak
idsysb
 Systems: 
 Cloak
idsysb
 Corolla and Explorer as separate  systems.
 Toggle Cloak
idintb
 Interactions: 
 Cloak
idintb
The impulse on each vehicle from the other is assumed to be the dominant interaction during the collision. Because we are now considering the vehicles separately, these are now external impulses.
 Toggle Cloak
idmodb
 Model: 
 Cloak
idmodb
.
 Toggle Cloak
idappb
 Approach: 
 Cloak
idappb
 Card
labelPart C
Part C
Assuming the collision lasted for 0.060 seconds, find the time-averaged force exerted on each vehicle.
Solution
 Toggle Cloak
idsysc
 Systems: 
 Cloak
idsysc
 Corolla and Explorer as separate  systems. 
 Toggle Cloak
idintc
 Interactions: 
 Cloak
idintc
 The external force on each vehicle from the other is assumed to be the dominant interaction during the collision.
 Toggle Cloak
idmodc
 Model: 
 Cloak
idmodc
.
 Toggle Cloak
idappc
 Approach: 
 Cloak
idappc
 Card
labelPart D
Part D
Suppose a 75 kg person in each vehicle underwent the same change in velocity as their automobile in the same amount of time. Find the time-averaged force exerted on these people.
 Toggle Cloak
idsysd
 Systems: 
 Cloak
idsysd
First, the Corolla and Explorer as separate  systems, then the passengers as separate  systems.
 Toggle Cloak
idintd
 Interactions: 
 Cloak
idintd
 Each vehicle is subject to a collision force from the other. The passengers are each subject to some decelerating force, possibly a seatbelt or airbag.
 Toggle Cloak
idmodd
 Model: 
 Cloak
idmodd
.
 Toggle Cloak
idappd
 Approach: 
 Cloak
idappd