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Kinematics question -- 2 runners racing with different acceleration rates

  1. Feb 11, 2015 #1
    1. The problem statement, all variables and given/known data
    Justin is racing Mark. Justin accelerates from rest at the starting point at 1.2 m/s^2 [E]. Mark is still adjusting his equipment and 2.0 s later begins to accelerate at 1.5 m/s^2 [E]. Where and when does Mark catch up to Justin?
    2. Relevant equations


    3. The attempt at a solution
    I think I need to create two formulas, set them equal to each other, and use the quadratic formula. I'm not sure how to create the formulas though.
     
  2. jcsd
  3. Feb 11, 2015 #2

    haruspex

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    What equations of motion under constant acceleration do you know? There are a standard five "SUVAT" equations.
     
  4. Feb 11, 2015 #3
    v=u+at, d=ut+1/2at^2, d=vt-1/2at^2
     
  5. Feb 11, 2015 #4

    haruspex

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    Ok. You can see that there are five variables that recur, two of speed, one acceleration, one distance, and one time. Which of those five is of no interest here? Find the equation that does not involve that variable.
     
  6. Feb 11, 2015 #5
    So d=ut+1/2at^2.
     
  7. Feb 11, 2015 #6

    haruspex

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    Right. So write out that equation for each runner, and the relationships between the times and distances in the two equations.
     
  8. Feb 11, 2015 #7
    How would I set them up if there are two unknown variables t and d?
     
  9. Feb 11, 2015 #8

    haruspex

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    There'll be two unknowns in each equation, but in each case there's a known relationship between the unknown in one equation and the corresponding unknown in the other equation. So you have enough equations to solve for all unknowns.
     
  10. Feb 11, 2015 #9
    Which variables are related?
     
  11. Feb 11, 2015 #10

    haruspex

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    The accelerations are known to be different, and the initial velocities are both known. What does that leave?
     
  12. Feb 11, 2015 #11
    Distance. But I don't think the equation would be d=0.6t^2 and d=0.75t^2 right?
     
  13. Feb 11, 2015 #12

    haruspex

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    That's right, because the the times are different. But they are related - how?
     
  14. Feb 11, 2015 #13
    The distance is the same.
     
  15. Feb 11, 2015 #14

    haruspex

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    Sure, but how are the two times related? You are told.
     
  16. Feb 11, 2015 #15
    Yes, one is two seconds longer, t+2.
     
  17. Feb 11, 2015 #16

    haruspex

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    Right. So what equations do you have now?
     
  18. Feb 11, 2015 #17
    I don't know where t+2 would go.
     
  19. Feb 11, 2015 #18

    haruspex

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    Simplest way to think about it is this: let t1 be the first runner's acceleration time and t2 be the second runner's acceleration time. That allows you to write the two SUVAT equations. Now you just have to relate t1 and t2 by the fact that they differ by 2 - the tricky part is to make sure you get it the right way round. Think: which runner runs for longer?
     
  20. Feb 11, 2015 #19
    I'm still not sure how I add the 2 seconds into the equation.
     
  21. Feb 11, 2015 #20

    haruspex

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    If Justin (first runner) runs for t1 seconds, how long does Mark run for?
     
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