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A pendulum is composed of a light rod with length $l$ and an end triangular head with mass $M$ which makes an angle $\alpha$ with the ground. The pendulum is released from an initial angle $\theta_0$ and flows freely. At its lowest position there is a small sphere with mass $m$ at rest. The triangle hits the sphere when it is at its lowest position and the collision is elastic. Find the velocity of the sphere after the collision.

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I have been trying to solve this problem for a while, and I can find the velocity of the triangle before the collision (simple conservation of energy), but I don't understand how to formulate the collision problem, as the sphere is supposed to have an initial velocity with a certain angle ($\alpha$, I presume), but I cannot understand why. Thank you for any help you may give me.

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At first sight it seems impossible to answer unless you know the speed of the wedge after the collision - then you'd use conservation of momentum.

If the question said that the wedge was stopped by the collision, then do conservation of momentum horizontally to find the horizontal component of velocity of the ball.

Since the ball will move initially perpendicular to the surface of the wedge, you'll also be able to puzzle out it's vertical component.


After question edit: It'll be a bit complicated but set up 3 equations with 3 unknowns: $v$ the final velocity (horizontal) of the wedge (the initial $u$ is now known), $x$ the horizontal velocity of the ball and $y$ the vertical velocity of the ball

  1. From conservation of energy $$M(u^2-v^2)=m(x^2+y^2)$$

  2. horizontal momentum $$M(u-v) = mx$$

  3. a third relating $x$ to $y$ to do the angle of the wedge and things mentioned earlier...best of luck!

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  • $\begingroup$ The collision is elastic. Sorry... I will edit the post adding it. $\endgroup$
    – Logos
    Jun 16, 2021 at 8:37
  • $\begingroup$ answer edited, over to you now! $\endgroup$ Jun 16, 2021 at 8:51
  • $\begingroup$ Thank you, it makes much more sense now. Can you please just clarify why the ball will initially move perpendicularly to the surface? $\endgroup$
    – Logos
    Jun 16, 2021 at 9:58
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    $\begingroup$ There will be a normal contact force in that direction. It's true there could also be a component of force parallel to the surface, but presumably we could ignore that (perhaps it mentions 'smooth'), if there was that component too, it could make the ball spin, but we would need to know the coefficient of friction, and other things... $\endgroup$ Jun 16, 2021 at 10:43

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