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Finding the radius of the sphere

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  1. Dec 28, 2017 #1
    1. The problem statement, all variables and given/known data
    In a spherical bowl a small ball is jumping around. The ball hits the bowl in two spots and moves in two different trajectories in times T1, T2. Find the radius of the bowl.

    sphere_image.png


    2. Relevant equations
    Newton's laws of dynamics.

    3. The attempt at a solution
    The only thing I could figure out is how the horizontal velocities of different trajectories are related:
    [tex] s=v_{x1}⋅T_1=v_{x2}⋅T_2 ⇒ \frac{v_{x1}}{v_{x2}} = \frac{T_2}{T_1} [/tex]
    But I can't find a way to get any expression with the radius. Please help.
     
  2. jcsd
  3. Dec 28, 2017 #2

    mfb

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    What do you know about the incoming and outgoing angle for a perfectly elastic collision?

    Unrelated to that: You will probably need the trajectories in both parts.
     
  4. Dec 28, 2017 #3
    Momentum is conserved. Then the velocities of both trajectories are equal. I still can't see how the radius is related to any of this.
     
  5. Dec 28, 2017 #4

    ehild

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    The ball interacts with the bowl which is much bigger than the ball. You can consider the bowl of infinite mass. The velocities are not the same, but energy might be conserved. What would it mean for the speeds?
    The ball performs two different projectile motions with the same range.
    upload_2017-12-28_14-28-36.png
     
    Last edited: Dec 28, 2017
  6. Dec 28, 2017 #5

    BvU

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    Are you saying the two different trajectories have the same path length ?
     
  7. Dec 28, 2017 #6
    Ok, so the velocities are not equal. I can’t really apply the momentum conservation law since the direction of motion changes after the impact, and I don’t have any angles.

    As for the energy I can only say that [tex]v_1<v_2[/tex] since the path [tex]T_1[/tex] has more potential energy (jumps higher).

    The trajectories have the same horizontal distance, which I denoted by s.
     
  8. Dec 28, 2017 #7

    ehild

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    What do you mean on v1 and v2?
    The problem said
    that means the ball repeats its motion, so the collisions can be considered elastic.
    The ball has both horizontal and vertical components of velocity. It jumps higher if the vertical component is bigger.
    T1 and T2 are the times between two collisions with the bowl.
     
  9. Dec 28, 2017 #8

    BvU

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    It's the directions, and the directions only, that change !
     
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