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GregA
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This is a problem that though I've solved has had me racking my brains trying to figure out whether or whether not I actually understand what I'm doing..here goes:
Particles A and B have mass m and are moving in the same direction along a line, A with speed 3u and B with speed u. They collide and after the impact they both move in the same direction, A with speed u and B with speed ku. The coefficient of restitution is e
a) show that e = (k-1)/2
b) Deduce that 1 =< k =< 3
c) Find the loss in kinetic energy in terms of m, k, and u.
As I said above...reaching the answers to these questions is not the problem...the bit that's bothering me is that before getting to the bit where I could solve part a) I had come to the conclusion that k can only = 3. My reasoning is as follows:
The total momentum of the system before impact is 3mu + mu, and given that I know the momentum of A after impact (mu) I can say that:
mu + kmu = 4mu...k = 3...the other way I look at it is as follows:
B imparts an impulse on A that changes it's velocity from being 3mu to mu, in doing this A must impart the same impulse to B such that:
-(mu - 3mu) = (kmu - mu)...k still = 3 (the minus sign is because J acts in different directions)
The question implies however that e and k are not already determined and can take any value beween 0,1... 1,3 respectively.
The only conclusion that I can reach however is that this is wrong and that given the mass of both objects, their initial speeds and either the value of e or just the speed of either A or B any other values are forced...this question stated otherwise
Particles A and B have mass m and are moving in the same direction along a line, A with speed 3u and B with speed u. They collide and after the impact they both move in the same direction, A with speed u and B with speed ku. The coefficient of restitution is e
a) show that e = (k-1)/2
b) Deduce that 1 =< k =< 3
c) Find the loss in kinetic energy in terms of m, k, and u.
As I said above...reaching the answers to these questions is not the problem...the bit that's bothering me is that before getting to the bit where I could solve part a) I had come to the conclusion that k can only = 3. My reasoning is as follows:
The total momentum of the system before impact is 3mu + mu, and given that I know the momentum of A after impact (mu) I can say that:
mu + kmu = 4mu...k = 3...the other way I look at it is as follows:
B imparts an impulse on A that changes it's velocity from being 3mu to mu, in doing this A must impart the same impulse to B such that:
-(mu - 3mu) = (kmu - mu)...k still = 3 (the minus sign is because J acts in different directions)
The question implies however that e and k are not already determined and can take any value beween 0,1... 1,3 respectively.
The only conclusion that I can reach however is that this is wrong and that given the mass of both objects, their initial speeds and either the value of e or just the speed of either A or B any other values are forced...this question stated otherwise
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(The question I have stated is quoted word for word from the book (forgot the period at the end of the paragraph though 
