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Elastic collision

  1. Jul 21, 2011 #1
    1. The problem statement, all variables and given/known data

    There is a ball rolling on a frictionless horizontal surface of mass m and with velocity 5m/s. It collides elastically with a block mass 3m that is initially hanging at rest from a 50 cm wire that is hanging from the ceiling.Find the maximum angle through which the block swings after it is hit

    2. Relevant equations

    v=initial velocity=5
    Rolling ball=m1=m
    Hanging block = m2=3m

    Velocity of hanging block after collision: (2m1/(m1+m2))v

    Velocity of rolling ball after collision: ((m1-m2)/(m1+m2))v

    3. The attempt at a solution

    So using the two equations above I get that the velocity of the block is 2.5m/s and velocity of the ball is -2.5m/s. So far this agrees with the fact that this is an elastic collision the knietic energy afterwards is equal to the knietic energy before the collision.

    So to find the angle I'm thinking that I need to find how far the block travels in the x direction, that way I can take the arctangent of the length of the string divided by the distance x to find an angle W.

    I use the work energy theorem:W=Fx=max=K after. The masses will cancel so I have ax=K after. But the acceleration is just dv/dt which is a change in velocity over a very short time interval. My thinking is the collision is short enough to be called dt and since it goes from rest to 2.5 m/s that a=2.5.

    X would then be (.5)(3)(2.5^2)/(2.5)=3.75 meters

    angle W would then be arctan(375/50)=82 degrees

    This does not seem reasonable since it would mean that the ball would leave the ground and some energy would be converted in to gravitation potential energy...help is appreciated
     
  2. jcsd
  3. Jul 21, 2011 #2

    gneill

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    Staff: Mentor

    The block WILL leave the surface, since it's suspended like a pendulum by the 50cm wire. That means energy WILL be converted into gravitational PE...
     
  4. Jul 21, 2011 #3
    How would I calculate the upward movement if the initial momentum is entirely in the x direction?
     
  5. Jul 21, 2011 #4

    gneill

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    Staff: Mentor

    Use a conservation of energy approach. You've already got the block's initial post-collision velocity...
     
  6. Jul 21, 2011 #5
    The initial kinetic energy was equal to 12.5 J and the kinetic energy of the block is 9.4 and the ball is 3.1 unless one of those velocities is wrong there isn't any more energy in the system to be accounted for...

    Also im curious, is the logic behind taking a=2.5 is sound?
     
  7. Jul 21, 2011 #6

    gneill

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    Staff: Mentor

    Since you don't have the actual masses of the ball or block, you can't really put a numerical KE value to either. (although you could say that the "specific KE" of the block is 12.5 J/kg)

    The KE of the block is going to be converted to gravitational PE as it swings upwards on the end of the wire.

    Regarding your value of acceleration, no, you really can't put a figure to the acceleration without knowing the actual time duration of the impact event. For nearly perfect undeformable objects this time would be vanishingly small, and the acceleration would approach infinite! Better to think in terms of impulse in those situations.

    But you've got the KE of the block, and a way forward with energy conservation. So onward-ho!
     
  8. Jul 21, 2011 #7
    Ok, So I set the KE=12.5=mgH and solved for H since this is where all the KE is converted into gravitational PE. H=.318. Since this is the distance above the ground I subtracted it from the length of the rope. This length then was the adjacent side of a right triangle formed by this side and the hypotenuse equal to the length of the string. arcos(50-32)/50 gave an angle of 67 degrees.
     
  9. Jul 21, 2011 #8

    gneill

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    Staff: Mentor

    Keep in mind that the mass of the block is 3m...
     
  10. Jul 21, 2011 #9
    My mistake KE=12.5 was supposed to be KE=9.4...I should write it out: .5mv^2=mgh ==> H=v^2/2g. v=2.5, So 2.5^2/2(9.8)=.32
     
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