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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For ideal, perfectly rigid objects it’s both, and that’s how we calculate their speeds after collision. Total energy is also always conserved. However, in real world scenarios some energy will be turned into heat by the collision while total momentum cannot go anywhere.
Momentum also takes into consideration vector, so it can describe the outcome of two objects colliding head on or at an angle.
Kinetic energy is not necessarily conserved. In fact, it is rarely conserved. In most collisions, a ton of energy changes to other forms, such as heat.
Momentum, however, cannot change forms to something else and continues to exist as momentum.
So no matter what, momentum must be conserved, and *sometimes* kinetic energy can be conserved.
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!
>Why do we say momentum is conserved, instead of total energy is conserved?
Total energy is conserved, however total **kinetic** energy is not always conserved. Energy can be transformed into other forms, if you throw two blobs of clay at each other then a large amount of their energy upon collision will go into deforming the blobs for example. Here kinetic energy has been transferred into the materials and will eventually end up as heat.
There are cases where the kinetic energy is also conserved, and we call these collisions elastic or perfectly elastic, and in those cases we can use both conservation of momentum and conservation of kinetic energy. In other cases we can still use kinetic energy to determine what happens, but we need to take into account how much energy gets lost.
Kinetic energy isn’t conserved because it can be transferred to other forms of energy. If two objects head straight towards each other and collide, bringing them to a stop, they start with a lot of kinetic energy and end with none.
Momentum, however, is conserved.
Mathematically the key difference is that kinetic energy comes from velocity squared, whereas momentum is just velocity.
Consider the example above. Two vehicles with equal velocity in opposite directions. So, one will have mv, one has – mv. If their masses are also equal, we can work out that their total momentum is 0, so after collision it will also be 0.
If you used kinetic energy, you wouldn’t get that, because the minus sign disappears when you square it, so they no longer sum to zero.
Basically, squaring velocity means you lose information about its direction
If you take two identical balls and throw them at each other at the same speed, the first ball has positive momentum, and the second ball, moving in the opposite direction, has negative momentum. When they collide the net momentum is still zero, meaning they exit the collision in opposite directions and moving the same speed. If they hit off center they can each go off at a right angle to their initial path.
Depending on the material properties of the ball though, some of that energy gets turned into heat. If you have two perfectly bouncy balls, they rebound at the same speed they started, but if you have two balls of dough that stick together, all that kinetic energy gets turned into heat. This is the difference between an elastic collision and an inelastic collision.
Kinetic energy can’t be conserved in a collision. Consider filming a head on collision between two hard spheres that are moving toward each other. The spheres each have mass and velocity and therefore positive kinetic energy. Kinetic energy is always positive because the direction of motion doesn’t matter as the velocity is squared. If you were to observe the balls after the collision they would be moving in opposite directions. But for an object to initially be moving in one direction and later be moving in the opposite direction, there must have been a moment when the velocity was zero. The same is true for the other sphere. At the moment of the collision the kinetic energy dropped to zero. This means that kinetic energy is not conserved. The energy didn’t vanish, it changed form. When the spheres are at rest, acoustic, elastic and thermal energies were present. The electric energy can be converted back into kinetic energy in the next moments as the spheres rebound, but the acoustic and thermal energy won’t be converted to kinetic energy. So even though the balls are moving after the collision, they have less kinetic energy than before the collision.
The solution to momentum being conserved is actually the solution to both equations.
m1v1 + m2v2 = m1v3 + m2v4 even if we know m1, m2, v1, v2, has infinite solutions because we only have 1 equation and 2 unknowns, so to.solve it we need 1/2(m1v1^2 + m2v2^2 ) = 1/2(m1v3^2 + m2v4^2). On its own, it also has infinite solutions, but with the momentum equation, we get exactly one solution (2 equations and 2 unknowns)
Usually when doing momentum conservation problems, the collision is either perfectly elastic (no energy is lost) resulting in both equations needing to be used. Or perfectly inelastic (maximum energy is lost) where the objects stick together m1v1 + m2v2 = (m1+m2)v3 which is just one equation and one unknown. Now that energy isn’t lost, but it’s given off as heat or sound or fragments of the objects breaking off and flying away. Now no collision is perfectly elastic or inelastic, so in the real world we would use m1v1 + m2v2 = m1v3 + m2v4 and/2(m1v1^2 + m2v2^2 ) = 1/2(m1v3^2 + m2v4^2) + H where H is the energy lost to heat in the collision. Of course, this gives us another unknown, which means there’s infinitely many solutions again. If we can predict the amount of heat that will be lost (based on the nature of the materials colliding) we can get a very good estimate of how the collision will play out.
So the answer is basically that we use both.
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