elastic vs. inelastic collisions

[epsilonprodigy]

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Ball A and ball B are suspended from strings. Ball A swings down in a semi-circle from height h, and hits ball B.
If ball A has a mass that is "much" larger than ball B, and hits a stationary ball B at velocity v, what is the maximum velocity ball B can attain, assuming a perfectly elastic collision?

• In other words, mass of A is approaching infinity, and mass of B is approaching zero.
• Kinetic energy will be fully transferred and conserved, because it is an elastic collision.
• If "all" of ball A's kinetic energy is transferred to ball B, ball A will then come to a stop.
• Ball B will then assume all of the kinetic energy that ball A had, most of which will be accounted for by its velocity, since mass of ball B is approaching zero and is insignificant in comparison to mass of ball A.

I did this with momentum and energy equations both, and got that technically in this case, the velocity of ball B would then also be approaching infinity. But the book says that the limit of ball B's velocity is 2v, or twice the velocity of ball A. I did it accounting for and ignoring gravity (by first equating ball A's potential energy with its kinetic energy and plugging sqrt 2gh in as velocity.) I always get that ball B's velocity would be approaching infinity since the momentum and KE of ball A were approaching infinity. I am going to bring it to a physics professor on Monday, but does anyone see anything wrong with my argument?[/QUOTE]



Someone had said this was incorrect, because if the collision is elastic, it is impossible for ball A to have stopped. Is this true, that is, if kinetic energy was completely transferred (and therefore conserved) then this somehow violates the conditions of a perfectly elastic collision? If so, how? (Simultaneously looking this up but sometimes it's better explained by an actual person.)

 

[ilovemcat]

Ball A and ball B are suspended from strings. Ball A swings down in a semi-circle from height h, and hits ball B.
If ball A has a mass that is "much" larger than ball B, and hits a stationary ball B at velocity v, what is the maximum velocity ball B can attain, assuming a perfectly elastic collision?

• In other words, mass of A is approaching infinity, and mass of B is approaching zero.
• Kinetic energy will be fully transferred and conserved, because it is an elastic collision.
• If "all" of ball A's kinetic energy is transferred to ball B, ball A will then come to a stop.
• Ball B will then assume all of the kinetic energy that ball A had, most of which will be accounted for by its velocity, since mass of ball B is approaching zero and is insignificant in comparison to mass of ball A.

I did this with momentum and energy equations both, and got that technically in this case, the velocity of ball B would then also be approaching infinity. But the book says that the limit of ball B's velocity is 2v, or twice the velocity of ball A. I did it accounting for and ignoring gravity (by first equating ball A's potential energy with its kinetic energy and plugging sqrt 2gh in as velocity.) I always get that ball B's velocity would be approaching infinity since the momentum and KE of ball A were approaching infinity. I am going to bring it to a physics professor on Monday, but does anyone see anything wrong with my argument?

For an ELASTIC Collision:

When a LARGE ball collides with a SMALL ball:
The velocity of the LARGE ball AFTER the collision will be between 0 m/s and V.
The velocity of the SMALL ball AFTER the collision will be between V and 2V.

Where "V" represents the (max) initial velocity of the LARGE ball at the bottom of the arc (just as it hits ball B).

So for this question, we're told that Ball A (Large Ball) collides with Ball B (Small Ball).

In order for momentum and KE to be conserved, the maximum velocity the small ball could attain is twice the initial velocity of the large ball. Truthfully, there's a mathematical approach you can take to prove this - but for the MCAT, this a fact you probably should just memorize (although this stuff is low-yield).

Because we're told the mass of A is "much" larger, then 2V is the best answer.

Analyze and understand the Graph EK provides for this chapter. These questions are testing you on those concepts.

 

[Rabolisk]

Just to be clear, it is impossible for ball A to stop after a perfectly elastic collision if the mass of ball A is greater than that of ball B. Any collision results in conservation of momentum. If ball A stops, then all of the momentum in ball A has been transferred to ball B. But then the kinetic energy of ball B after the collision will not equal that of ball A before the collision unless their masses are equal. Mathematically it is impossible. Mathematically it can also be shown that the limit of ball B is 2v.