I think i’m missing something. When two object is colliding and no energy is lost why is the outcome determined by the total momentum formula? (m1v1 +m2v2 = m1v1′ + m2v2′)
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Why is it not total kinetic energy ? (¹/² m1.v1² + ¹/² m2.v2² = ¹/² m1.v1²’ + ¹/² m2.v2²’)
Why do we say momentum is conserved, instead of total energy is conserved?
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We can understand this problem using three equations.
F = m * a — force = mass times acceleration
p = m * v — change of momentum = mass times change of velocity velocity
v = a * t — change of velocity = acceleration times change of time
Combining the second two equations gives us:
p = m * (a * t) — change in momentum = mass times acceleration times change of time
Then we put the first equation in there:
p = (F )* t — change in momentum = force times change of time
From Newton’s third law we know that every action has an equal and opposite reaction, so during the length of time that the objects are colliding they must be exerting equal and opposite forces on each other. This will (looking at the whole system) make F = 0. So, change in momentum of the system = 0.
Looking at each individual object, lets say the force is 10 newtons. The other object will exert 10 newtons in the opposite direction, so this is -10 newtons. Thus, the change in momentum of one will equal the opposite of the change in momentum of the other.
This does not work for kinetic energy (try it yourself and you’ll see). Beyond the math, we can imagine what happens to the energy in the system. Two object collide with a fixed amount of energy – some of this energy turns into sound, some is used deforming the objects, some it turned into thermal energy (smack a nail with a hammer a few times and you’ll feel the heat on the nail). The kinetic energy has some place to “go”, while the momentum MUST be equal and opposite so it has nowhere to “go”.
Hope this helps!
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