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Can you solve this collisions problem?

So, you've got a central elastic collision between two balls. The two balls have opposite momentums. Prove that the kinetic energy of ball 1 before the collision is the same as the kinetic energy of ball 1 after the collision.
Elastic collision means the energy of the system is conserved. So, the net energy before and after the collision is the same. You mentioned that the objects have opposite momenta. I assume that this means the momenta are equal (Please get back to me ASAP if it's not... I may be missing something here :smile: )

Therefore,
Initial momentum = Final momentum
m1u1 + m2u2 = m1v1 + m2v2
But, both momenta are equal.
=> m1u1 = m2u2
Thus, m1u1 = m1v1

As far as I know, this should be right. However, do not hesitate to offer any suggestions, changes, corrections, etc. :smile:

~Ri
Reply 2
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Stef_11
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There's a problem with your solution. Momentum is a vector quantity and in your equation you didn't use the direction of the vectors. The correct equation should be m1u1 - m2u2= -m1v1 + m2v2. Which doesn't really lead somewhere.
Original post by Stef_11
There's a problem with your solution. Momentum is a vector quantity and in your equation you didn't use the direction of the vectors. The correct equation should be m1u1 - m2u2= -m1v1 + m2v2. Which doesn't really lead somewhere.


Are these balls identical? (not stated in OP)
Original post by Stef_11
So, you've got a central elastic collision between two balls. The two balls have opposite momentums. Prove that the kinetic energy of ball 1 before the collision is the same as the kinetic energy of ball 1 after the collision.


You need to give more details of the scenario, i.e whether two balls have a velocity or if they are of equal mass. The only way that the balls can have opposite momentum is in a head on collision.
Reply 5
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Stef_11
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Guys, no data is given about the mass of the balls. I already mentioned that the collsion is head-on ( I said it is a CENTRAL collision) . This problem is an actual problem from a physics book, which doesn't include the answer. I didn't make up the problem. Your guess about the balls is as good as mine.
Original post by Stef_11
Guys, no data is given about the mass of the balls. I already mentioned that the collsion is head-on ( I said it is a CENTRAL collision) . This problem is an actual problem from a physics book, which doesn't include the answer. I didn't make up the problem. Your guess about the balls is as good as mine.


If the balls are of equal mass then you would know that after the collision the balls will travel with the same speed in opposite directions to their original velocity (because they must have the same velocity to have equal and opposite momentum). In this case, you can easily deduce that the initial kinetic energy is equal to the final K.E. However, if the balls have different masses then they must have different velocities to have equal and opposite momentum but proving the conservation of K.E would be something of a challenge but essentially rooted in algebraic manipulation.
Reply 7
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Stef_11
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If the balls are of equal mass, I need to prove it. If they don't, I still need to prove that V1=v1' and yes, I did use mainly algebraic manipulation when I attempted to prove the latter, however, with no result. Basically, I was only getting relationships that I already knew to be true.
Original post by Stef_11
If the balls are of equal mass, I need to prove it. If they don't, I still need to prove that V1=v1' and yes, I did use mainly algebraic manipulation when I attempted to prove the latter, however, with no result. Basically, I was only getting relationships that I already knew to be true.


I think you need to at least show that u1=v1 if that's what you mean but, thinking about it again, I don't think we could prove this mathematically. The idea of K.E being conserved in an elastic collision is assumed to be true on the basis of the concept of conservation of energy.
Reply 9
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Stef_11
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But KE being conserved applies only for the KE of the system before an after the collision, not for individual KEs.
Original post by Stef_11
But KE being conserved applies only for the KE of the system before an after the collision, not for individual KEs.


That's true, and in fact this is not always the case, but in the case of two bodies of equal mass making a head on collision with each other with the same velocity we know that, according to Newton's third law, they exert forces of equal in magnitude but opposite in direction on each other. Because they are of equal mass, they must travel back with the same velocity in opposite direction to their original velocity.
(edited 8 years ago)
Original post by Stef_11
There's a problem with your solution. Momentum is a vector quantity and in your equation you didn't use the direction of the vectors. The correct equation should be m1u1 - m2u2= -m1v1 + m2v2. Which doesn't really lead somewhere.


....The equation you mentioned wasn't useful, which is why I tried it the other way... But yeah, that was pointless :facepalm:
Reply 12
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Stef_11
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We don't know the bodies are of equal mass. 😔
Original post by Stef_11
We don't know the bodies are of equal mass. 😔


Even if they the masses are not equal but their momentum is opposite to each other, still the conservation of K.E (of individual balls or of the system) is an assumption made on basis of the Energy conservation law.
His other thread is here:
http://www.thestudentroom.co.uk/showthread.php?t=3736243

Original post by Stef_11
I mentioned that the momentums of the balls are opposite i.e they have the same magnitude and opposite directions. I didn't mention that the momentums are opposite INITIALLY, my bad. Anyway, in all my efforts I have made use of the following relationships: m1v1 - m2v2= -m1v1' + m2v2' and KE1 + KE2=KE1' + KE2'. I also used p1=p2 and p1'=p2'. My goal is basically to prove that V1=v1'. I used the above relationships in many different ways, solving and substituting but that doesn't lead me anywhere, only to relationships I know to be true. Any ideas?


You should quote someone if you want a reply. I just saw this thread and checked your other one.

I think you're confused on conservation of momentum. If the initial momentums are equal and opposite, what is the total initial momentum?
Original post by morgan8002
His other thread is here:
http://www.thestudentroom.co.uk/showthread.php?t=3736243



You should quote someone if you want a reply. I just saw this thread and checked your other one.

I think you're confused on conservation of momentum. If the initial momentums are equal and opposite, what is the total initial momentum?


I think he's taken away the initial momentum to take the direction of velocities into account. How would you go about proving the conservation of K.E by considering the total initial momentum to be zero?
Original post by Mehrdad jafari
I think he's taken away the initial momentum to take the direction of velocities into account. How would you go about proving the conservation of K.E by considering the total initial momentum to be zero?


Conservation of kinetic energy is given in the question(elastic).
There are three pieces of information: conservation of kinetic energy, conservation of momentum, equal and opposite initial momentum. I did it by writing the last two of these in equation form, combining them and substituting into the first.
Original post by morgan8002
Conservation of kinetic energy is given in the question(elastic).
There are three pieces of information: conservation of kinetic energy, conservation of momentum, equal and opposite initial momentum. I did it by writing the last two of these in equation form, combining them and substituting into the first.


True, elastic is a key word here but I couldn't show it mathematically. Although I did find the kinetic energy of the system initially in terms of the final kinetic energy, I couldn't see how they were equal.
Original post by Mehrdad jafari
True, elastic is a key word here but I couldn't show it mathematically. Although I did find the kinetic energy of the system initially in terms of the final kinetic energy, I couldn't see how they were equal.


Elastic means the final kinetic energy of the system is equal to the initial. That's the definition of elastic.

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