Understanding Impulse and Energy Transfer in Elastic Collisions

[Sefrez]

I have been messing around with a metal ruler, a slick surface, and a small flat head screw driver at which I have inserted into a plastic tube allowing me to connect a rubber band. It is basically a "pinball" launcher, if you will. I did this to create an impulse on the ruler at a high force over a little time interval to reduce effects of friction.

I know that when a force is applied to an object that no matter if there is a torque or not, the center of mass accelerates as if the force was applied there. But what about impulse between two bodies in an elastic collision? I figure the same stands, but the time and how the force is applied (I realize that in a collision this can be complex) must change depending on where the two bodies collide.

That is, doesn't a body that has translational motion only have less than that same body also having rotation motion? (E.g. k = 0.5mv^2 + 0.5Iw^2 > 0.5mv^2)

If that is true and it is true that the center of mass accelerates as if a force was applied directly to it - then that means a mass will have a total energy greater when a force also caused a torque upon it.

This makes sense in itself, but in an elastic collision where you have impulse (where forces may be more complex), it must be different as energy must be conserved. That is, if A object collides with the center of mass object B, B should translate greater than if it were to have been collided with its edge to conserve energy. Either that, or object A must retain more energy when colliding with the center of mass B to account for the rotational energy that B does not have.

Is this not correct?

Going back to my experiment with the ruler, I could not get any very accurate results, but I did seem to notice that the ruler translated about the same no matter where I hit it. The only thing that seemed to change much is angular rotation. My confusion if the results are valid? What I have just stated above and also what seems pretty clear to me - that the rubber band should have a quantifiable potential energy when pulled back to point x and thus the ruler should not have more total energy when hit at the edge. So my only assumption to this is that the amount of energy transfer between the two must vary depending on the point of contact. Is this correct?

 

[AlephZero]

Mechanical energy is only conserved in a perfectly elastic collision. Hitting a metal ruler with an elastic band is not a good approximation to perfectly elastic. Hitting the ruler with a metal ball bearing would be closer, if you want to try that. You will then have to estimate the energy remaining in the ball bearing after it bounces off the ruler.

Your basic idea that the translational velocity of the ruler's center of mass is constant but the rotational velocity depends on the position of the impact is correct. The quantities that are conserved (exactly, in any type of collision) are linear and angular momentum. The intial angular momentum of the system about the center of mass of the ruler depends on the position of the elastic band.

You are right that the amount of mechanical energy transferred depends on the point of impact. An experiment to show that would be to fix one end of the ruler so it can vibrate like a beam. Impact it at different points along its length from the fixed end to the free end, and see how the amplitude of vibration changes.

This sort of observation is good "real engineering". These issues are imprtant for understanding what happens to real-world structures and machines, but often they get forgotten, or they are hidden by pages of equations and/or computer graphics.

 

[Sefrez]

Thanks for the reply! :) Yeah, I would like to code a simple program (finding it not so simple) where I have to masses collide (2D), but I am finding it very difficult to calculate everything that goes on. I am assuming there are better tricks to go about doing this though. Like calculating angular momentum about different points - especially where they are not constant in time. This "simple" kind of collision is driving me nuts! **thinks how a human could ever had created a physics engine**