# conservation of momentum and energies in collisions

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hello, ok this is confusing me a lot, i will try and explain my question as best as possible. ok say i have an object called A who's mass is 10 kg and it's velocity is 10,000 m/s p = mv so it's momentum would be 100,000 Kgm/2 and Ek = (1/5)mv^2 so it's kinetic energy would be 5*10^8 J now if i have object B that has a mass of 10,000 kg and a velocity of 10 m/s p = mv so B will have same momentum of A of 100,000 Kgm/s and Ek = (1/5)mv^2 so it's kinetic energy would be 5*10^5 J ok so they have the same momentum but the faster moving object has a lot more energy ... so this means that object A on contact with another object can do more work on that object than if object B comes into contact with the same object. becasue of course W = FD so the one with more energy or "work" (object A) most exert a greater force and move something a greater distance than that of object B. ok here is my question, if A has more energy than B but the same momentum, does the force on the object increase or the distance? but then i thought if they have the same momentum the velocity of the object A or B hits is the same, which means the force A exerts is = to the force B exerts, so does only the distance increase? so as the objects kinetic energy raises only the distance that objects can move something raises and the force remeans the same in this type of situation? it all hurts my brain. i hope some of this made sense in any way, thanks for any help in advance. EDIT: wow i did write 1/5 mv^2 i ment 1/2 i guess i was thinking of 0.5 and the fraction form at one time [Edited by - ekrax on October 5, 2004 9:41:08 PM]

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First of all, it's 1/2 MV^2 (not 1/5, but that was probably just a typo.)

Now, when either of the bodies strikes something, they will impart the same amount of force (f=ma) and, I believe, exchange the same amount of momentum.

That said, I must admit I don't understand the full meaning/applicaton of Ek in this sense. Perhaps it is because impacts are often modeled as instantaneous events (correctly or incorrectly) and thus the time is not a factor and talking about "work" doesn't apply.

Somebody with good physics knowledge should chime in.

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When you only conserve momentum it is ambiguous what object has what velocity. That is part of the reason why conservation of energy is used, because when both equations are solved simultaniously you can find each velocity.

Of course, KE is only conserved in very strict circumstances, ie the elastic collision.

As far as which crestes more force, I'm not sure what to say in a general case that doesn't have too many assumptions. All I can think to tell you is that the true definition of Force is -dp/dt, and not F=ma.

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Quote:
 Original post by ekraxok here is my question, if A has more energy than B but the same momentum, does the force on the object increase or the distance?

Both can happen. E.g. consider A and B being brought to a halt by a constant force. As momentum = Force x Time they take the same time to stop, but A which has higher speed goes further in that time. Work done = Force x Distance so A does more work.

Alternately consder A and B hitting a viscous liquid that stops them in the same distance (this could be done e.g. by shaping them to provide enough drag). As A is travelling faster it needs to decelerate much more sharply to stop in the same distance, so the force on it and the force it exerts on the liquid (equal and opposite) must be greater.

(The equation relating velocity and acceleration/deceleration is
v^2 = u^2 + 2ax
which shows how acceleration scales with the square of the initial speed in this case.)

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Actually, in an elastic collision, a smaller object will impart more energy to an object it colides with than a large one with the same momentum. Example:

A 1 kg ball hits another 1kg ball at 100 m/s. The collision is elastic, so the first ball stops and the other shoots off at 100 m/s.
Then, a 100kg ball going 1 m/s hits the same 1 kg ball. I'm sure you can see just by visualising this that the 1 kg ball will not shoot off at 100 m/s, but will instead move much slower. Obviosly, a ball with equal mass moving more slowly has less energy, thus the larger ball imparted less energy.

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this somewhat makes sense in that Force x Time = Momentum so
that the one going faster goes further in the same time so it does more work. and the example about the balls stopping in water makes sense also.

ok so you can generally say that if two objects have the same momentum but different velocites, the one with more velocity can do more work becasue it can travel a greater distance in the same time as the slower object. and that object can exert a greater force if stopped the same time as the other objects.

but can 2 objects have the same momentum but the hevier one does more work? i guess not now that i think about it since if one was the size of the earth moving 1 m/s and the other was 1 kg and moving insanly fast no matter what the velocity is sqaured and will always be bigger than the mass ok yeah.

thanks for the replies, if anyone has anything more to add i would be happy to read it.

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It might be clearer if you rearrange equations so they involve the things you are interested in.

E.g. kintic energy, and so work done stoppng something, is

E = 1/2mv^2

But momentum is

p = mv

So

v = p / m

and therefore

E = 1/2 m (p/m)^2

= p^2 / (2m)

And it's now clearer that energy/work done is inversely proportional to mass if the momentum is fixed. Similar rearrangement and simplification can be used to find the connections between other quantities.

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