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I collided with confusion!

I am not able to understand collisions. I have tried learning from Irodov but I couldn't understand a non-central collision - how to find the solution. The vector diagram? That came out of nowhere. Can anyone provide me with a better insight into this?

I will try learning the same from Kleppner but still if anyone can help me here, it would be good.

Note by Kartik Sharma
2 years ago

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@Kartik Sharma Sorry for deviating completely, but which book of Irodov's are you talking about? I just know of 'Problems in General Physics'. Also, could you tell me about the Kleppner book you mentioned? I'd really love to have a look at some online versions of the books you mentioned. Ishan Dasgupta Samarendra · 2 years ago

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@Ishan Dasgupta Samarendra Irodov - Fundamental Laws of Mechanics

This one is my favorite - Kleppner An Introduction to Mechanics

I have provided you with the pdf links, just in case. Kartik Sharma · 2 years ago

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How to solve a collision problem does depend on the details, such as the geometry of the objects colliding and the point of contact, etc. Worse, if inelastic collisions are involved. But let's say that we only have "balls" (or circles in 2D) colliding. Then the most important thing to keep an eye on to keep from getting totally confused is that momentum is conserved along the line that is normal to the point of contact, i.e.,normal to the common tangent. And you only consider one such contact "at a time". If there are multiple simultaneous contacts, then we can have indeterminacy of outcome, i.e., it cannot be uniquely decided. Think of the break shot in pool. If the balls were separated slightly, so that there is only one contact at a time, then the result is deterministic.

Other than that, the usual conservation of kinetic energy, which is always in the direction of the travel of the objects relative to some reference frame, and conservation of momentum, which is true in any direction in space. For example, momentum is conserved in the x, y, and z axes independently. This is where momentum, being a vector and not a scalar like kinetic energy, can be decomposed at center of mass of objects into momentum along that normal, and into momentum perpendicular to it. Then "momentum is conserved" along that normal, before and after collision. In this way, you basically reduce a 2D or even a 3D dynamic situation of colliding balls (or circles) into 1D toy versions at each point of contact, where equations are much easier to handle.

It might interest you that Issac Newton did not come up with the concept of energy or kinetic energy, and yet he was able to work out planetary orbits purely on forces and momentum considerations alone. Using the conservation of kinetic energy just makes things easier to work out, but is not strictly necessary. The moral of the story is: keep an eye on those momentum vectors. Michael Mendrin · 2 years ago

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@Michael Mendrin The tricks in Center-of-mass system are quite un-guessable. And that was what I was asking how to observe. In particular, I saw a trick of making an arc kinda thing in C-frame to find the final velocities. This was what I was asking- what are they? where did that come from? Kartik Sharma · 2 years ago

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@Kartik Sharma For me to help you, I need a better idea of what you're asking about. Is there a particular problem that's stumping you?

By the way, those books by Irodov and Kleppner are quite nice, but there's many more you can find for free online. But since you're familiar with these, how about if you gave me a page number so I'll know what's confusing you? Michael Mendrin · 2 years ago

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@Michael Mendrin Nice idea!

Irodov - Pg. No. 131 (I mean from Page No. 131 I couldn't continue reading. You can see some "vector diagram of momenta". I didn't seem to get what does that mean)

Kleppner - Well, I will read the same in Kleppner too but I have not read it yet. Since, I am reading Chapter 4-5, I will be there in Chapter 6(where collisions are discussed) by tomorrow.

I guess you might have got some idea about what's confusing me. The vector diagram of collision, in particular. Kartik Sharma · 2 years ago

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@Kartik Sharma Okay, the book makes a good point about value of using the "C" or center-of-mass frame. We can assume that the center of mass of the system of 2 particles is stationary, so we only need to look at the particles from the point of view of C. In particular, if one particle is at rest and the other is moving, then relative to C, momentum is [still] conserved. That is, if m, M are the masses of the particles, and v, V are the velocities of the particles relative to C, then mv = MV. This diagram (in thin vectors) shows both moving relative to C, which is actually point O in the diagram (which might be why you got confused). After collision, again momentum is conserved in the same way, but at a different angle. With point-like particles, we don't know what that angle can be, because that depends exactly how the almost point-like particles strike--head-on, or glancing, or something in-between. Nevertheless, there is a relation between the angles of particles leaving the point of collision, as exemplified by a circular arc.

The conservation of energy forces an unique outcome given an angle of scattering relative to C,...er, O in this diagram. You could solve this uniquely without invoking the conservation of energy, but it's an extra hassle.

The vector diagram shows how this can be "seen" from both from C (or O) reference frame and the original stationary particle's reference frame. The thin vectors is what it looks like from C (or O) reference frame, and the thick vectors is what it looks like from the original stationary particle's reference frame. Either way, momentum is always conserved in any particular direction, and you have a choice of reference frame in which to analyze a problem like this.

The beauty of physics is that for a given problem like this, there are all kinds of ways to come up with the same result, so I would strongly suggest that you independently come up with the same conclusions that is not necessarily exactly the way the book presents it. If you plan to excel in physics, make it a habit to find other ways of getting the same results. Michael Mendrin · 2 years ago

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