183_notes:examples:two_students_colliding

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183_notes:examples:two_students_colliding [2014/10/01 05:01] – pwirving183_notes:examples:two_students_colliding [2014/10/02 15:50] (current) – caballero
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-===== Example: Student Collision Example. =====+===== Example: Colliding Students =====
  
 Two students are running to make it to class. They turn a corner and collide; coming to a complete stop. What force did they exert on each other. Two students are running to make it to class. They turn a corner and collide; coming to a complete stop. What force did they exert on each other.
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 ${p}_{fx} = {p}_{ix} + {F}_{x,coll} \Delta t$ ${p}_{fx} = {p}_{ix} + {F}_{x,coll} \Delta t$
 +
 +We are told that the students come to a complete stop and so ${p}_{fx}$ = 0.
  
 $0 = M{V}_{ix} + {F}_{x,coll} \Delta t$ $0 = M{V}_{ix} + {F}_{x,coll} \Delta t$
 +
 +Rearranging the equation we relate ${F}_{x,coll}$ to the remaining variables. We are trying to find force.
  
 $ {F}_{x,coll} = \dfrac{-M{V}_{ix}}{\Delta t}$  $ {F}_{x,coll} = \dfrac{-M{V}_{ix}}{\Delta t}$ 
  
-negative means $-\hat{x}$ direction+The negative sign means that the force is in $-\hat{x}$ direction.
  
-Need to find collision time ${\Delta t}$+Now that we have the above relationship we must find the missing variables in order to solve for $ {F}_{x,coll}$. First we need to find collision time ${\Delta t}$. We can relate the average velocity to displacement over time.
  
 $\vec{v}_{avg} = \dfrac{\vec{v}_{f} + \vec{v}_{i}}{2} = \dfrac{\Delta \vec{r}}{\Delta t}$ $\vec{v}_{avg} = \dfrac{\vec{v}_{f} + \vec{v}_{i}}{2} = \dfrac{\Delta \vec{r}}{\Delta t}$
  
-In 1D: $\vec{v}_{avg} = \dfrac{\Delta x}{\Delta t} = \dfrac{\vec{v}_{f} + \vec{v}_{i}}{2}$+ 
 + 
 +In 1D this looks like: $\vec{v}_{avg} = \dfrac{\Delta x}{\Delta t} = \dfrac{\vec{v}_{f} + \vec{v}_{i}}{2}$ 
 + 
 +Relate these 3 equations together to solve for ${\Delta t}$
  
 $ {\Delta t} = \dfrac{\Delta x}{{V}_{avg}} = \dfrac{\Delta x}{\dfrac{\vec{v}_{f} + \vec{v}_{i}}{2}} $ $ {\Delta t} = \dfrac{\Delta x}{{V}_{avg}} = \dfrac{\Delta x}{\dfrac{\vec{v}_{f} + \vec{v}_{i}}{2}} $
 +
 +Fill in the values for the variables from the assumptions and approximations you made previous.
  
 $ \dfrac{0.025}{\dfrac{5m/s + 0m/s}{2}} = 0.01s $ $ \dfrac{0.025}{\dfrac{5m/s + 0m/s}{2}} = 0.01s $
 +
 +Having solved for $ {\Delta t}$ fill this value and the known value for mass and the approximated value for velocity into the equation that we arranged earlier to find $ {F}_{x,coll}$
  
 $ {F}_{x,coll} = \dfrac{-M{V}_{ix}}{\Delta t} = - {\dfrac{(68kg)(5m/s)}{0.01s}}$ $ {F}_{x,coll} = \dfrac{-M{V}_{ix}}{\Delta t} = - {\dfrac{(68kg)(5m/s)}{0.01s}}$
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  • Last modified: 2014/10/01 05:01
  • by pwirving