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Two rocks launched up/collison (Kinematics)

  1. Sep 27, 2015 #1
    1. The problem statement, all variables and given/known data
    This is a quiz question that was given in my AP physics C class. "A rock is launched upward with a speed V0. With what velocity must a second rock be launched from the same initial position at time T later so that it hits the first rock when it has a velocity of -V0/2."

    2. Relevant equations
    The only formulas we needed for this question were the basic velocity, displacement, and acceleration formulas (as functions of time) and their variations. (1D kinematics).

    3. The attempt at a solution
    1) I drew a diagram on an axis with a vertical line (y axis) and the origin being zero (the initial position of the balls).

    2) I listed all of the knowns and unknowns given from the question. We know that:
    • Rock1 (first rock) has an initial velocity of V0.
    • Time= T
    • Acceleration= -g= -9.81 m/s2
    • When the second rock hits the first rock, the velocity of the first rock will be -V0/2.
    We don't know the velocity that is necessary for rock2 (second rock), which allows the given condition ( When the second rock hits the first rock, the velocity of the first rock will be -V0/2.) to happen. Basically, we don't know the initial velocity of rock2.

    I then assumed the conditions for the collision:
    When the rocks collide, they will be the same distance from the ground, so d1=d2
    and I also assumed the time of travel for both rocks at the collision, which I put as T=T, which is where I'm sure that I made a mistake.

    Since this is from a collected quiz, I don't remember exactly what I did. However, I remember doing something similar to this:

    (-V0rock1 /2)T- (1/2)gT2=( V0rock2)T- (1/2)gT2
    I forgot the process, but my solution was something like V0/2T, which is incorrect because the units come out to be m/s2. I feel that I did something completely off track, and I really need help as to how I should approach these problems and how to think more effectively. I felt as though this question should have been easier for me, which kind of discouraged me after that quiz was collected. This isn't the first time we get a question like this, and help for the question and questions like these would be much appreciated. Thank you in advance.
     
    Last edited: Sep 27, 2015
  2. jcsd
  3. Sep 27, 2015 #2

    haruspex

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    You seem to be confused about T. The given T is the interval between launch times. If the first rock is in flight for time t1 before the collision, how long is the second rock in flight until then?
     
  4. Sep 27, 2015 #3

    mfb

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    That is not a good idea if more than one time is involved. Labels like t1 and so on make it easier to see what happens when.
     
  5. Sep 27, 2015 #4
    Oh. I see what's going on. So I drew a graph just to help me visualize, where I assumed arbitrary times, where the first rock is thrown at 1s and the second is thrown at 4s. The time interval between them is 3s. This helped me confirm the equation that T=t2- t1, which means that t2=t1+T
     
  6. Sep 27, 2015 #5

    haruspex

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    Maybe. It depends how you are defining t1 and t2. Are these times at which the rocks are thrown, or the times they spend in the air?
     
  7. Sep 27, 2015 #6
    I defined them as the times that the rocks were thrown.
     
  8. Sep 27, 2015 #7
    Ok. I still assumed that d1=d2.

    V01t1+(1/2)gt12= V02(T+t1)+(1/2)g(T+t1)2
     
  9. Sep 27, 2015 #8

    mfb

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    Now you gave the second object more time of flight than the first one.

    Also, your use of t1 doesn't match the definition above, as it seems to be the total flight time of the first object here.
     
  10. Sep 28, 2015 #9

    haruspex

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    I'm still unclear what you mean. The times at which they were thrown or for which they were aloft?
    If they are the times at which the rocks are thrown, you might as well choose t1=0, making t2=T. But then you need another variable to represent the time of the collision.
    If they are the times the rocks spend in the air, which one should be the greater?
     
  11. Sep 28, 2015 #10
    Ok I think it's becoming clearer. I was going off of your first reply that T is the time interval between the launching of the rocks, and may have gotten confused from that. I now understand that T is the time in which they collide. Clearly, Rock1 is in the air longer because it was thrown first. Therefore, t1 is greater than t2. Therefore, the equation, T-t1=t2 must be true.
     
  12. Sep 28, 2015 #11

    haruspex

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    No, T is the time interval between launches. Reread the question.
    What relationship does that give you between the times they spend in the air, t1 and t2?
     
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