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Kinematics of three runners in a race

  1. May 25, 2013 #1
    1. The problem statement, all variables and given/known data

    Runners A, B, and C run a 100-m race, each at a constant speed.
    Runner A takes first place, beating runner B by 10 m. Runner
    B takes second place, beating runner C by 10 m. By what time
    interval does runner A beat runner C?

    2. Relevant equations
    d = 100 m
    Δd = 10 m
    Δt[itex]_{i}[/itex] = time for runner i to travel 100 m
    v[itex]_{i}[/itex] = speed of runner i

    3. The attempt at a solution
    Can this problem be solved without knowing the speed of one of the runners? Here's a few equalities for this problem:

    v[itex]_{a}[/itex] = [itex]\frac{d}{Δt_{a}}[/itex]
    v[itex]_{b}[/itex] = [itex]\frac{d-Δd}{Δt_{a}}[/itex]
    v[itex]_{c}[/itex] = [itex]\frac{v_{b}Δt_{b}-Δd}{Δt_{b}}[/itex]
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 25, 2013 #2
    Try assuming a couple of different velocities for runner A and then work out the time difference. Does the time difference vary with the velocity of runner A? This should answer your question.
     
  4. May 25, 2013 #3
    Note that runner A beats runner C by more than 20 m.
     
  5. May 25, 2013 #4

    rude man

    User Avatar
    Homework Helper
    Gold Member

    Yikes, that's right!

    I am deleting my post accordingly to think more about this.
     
  6. May 25, 2013 #5
    I tried assuming a few speeds for runner A and found that the interval between runners A and C depends on the speed of runner A. In other words, there is no unique solution with the given information.
     
  7. May 25, 2013 #6
    The problem has six unknowns (3 speeds and 3 times) and five equations. It is fairly trivial to express any five unknowns in terms of the other one.
     
  8. May 25, 2013 #7

    rude man

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    Homework Helper
    Gold Member

    Right all around. At least I got the 'no unique solution' part right ...
     
  9. May 27, 2013 #8
    time for C

    A beats C by a time period of 20/VC

    VC=velocity of C
     
  10. May 27, 2013 #9
    This assumes that A beats C by 20 meters. This is not given. What is given is that B beats C by 10 meters, which happens when B is at the finish line. When A is at the finish line, B is not there, so you can't assume that he is 10 meters ahead of C yet.
     
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