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Kinematics Problem and stream of people

  1. Feb 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Figure 2-21 shows a general situation
    in which a stream of people attempt to escape through an exit door
    that turns out to be locked. The people move toward the door at
    speed vs  3.50 m/s, are each d  0.25 m in depth, and are separated
    by L  1.75 m.The arrangement in Fig. 2-21 occurs at time t  0. (a)
    At what average rate does the layer of people at the door increase?
    (b) At what time does the layer’s depth reach 5.0 m? (The answers
    reveal how quickly such a situation becomes dangerous.)


    2. Relevant equations



    3. The attempt at a solution
    I am not certain how to solve this.
     

    Attached Files:

  2. jcsd
  3. Feb 27, 2012 #2

    gneill

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    Staff: Mentor

    What equations in general do you think are applicable?

    How long do you think the time interval will be between successive impacts?
     
  4. Feb 27, 2012 #3
    Perhaps t = x/v would be one of the equations. I know the first person impacts the wall in .5 seconds.
     
  5. Feb 27, 2012 #4

    gneill

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    Staff: Mentor

    Okay, so suppose the first person has hit the door and come to a halt. How long till the next impact occurs?
     
  6. Feb 27, 2012 #5
    Would it be another .5 seconds?
     
  7. Feb 27, 2012 #6

    gneill

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    Staff: Mentor

    Is that a guess, or do you have some reasoning behind that value?
     
  8. Feb 27, 2012 #7
    Well, I believe it would be another .5 seconds because they all have to travel the same distance--that is, L.
     
  9. Feb 27, 2012 #8

    gneill

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    Staff: Mentor

    Okay, so if the guy in front stops, the guy following has distance L to travel before impacting. Good.

    So, what can you say about the rate of impacts? Can you turn that into an average rate that the layer of people will increase?
     
  10. Feb 27, 2012 #9
    So, in 1 second the first person collides into the door, and then the second person collides into the first person. Since each person has a depth of .25 m, that would mean the depth at the door would increase .5 m/s? That doesn't seem right. Isn't the depth at the door increases exponentially?
     
  11. Feb 27, 2012 #10

    gneill

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    Staff: Mentor

    As you stated earlier, the first person hits the door in 0.5 seconds, not 1 second. After that they "pile on" at a rate of one every 0.5 seconds. At the moment you're interested in this "after that" scenario, since it represents the ongoing rate of accumulation of people on the pile.

    Note that there is no exponential increase in the rate that people arrive --- they arrive at a constant rate, so the depth is increasing at an average constant rate, too: At 0.5m/sec in fact, as you've discovered.

    You're now in a position to answer part (b).
     
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