I'm trying to explain to someone learning elementary physics (16 year old) that linear momentum and energy are conserved independently in a 2-body collision. I'm not a professional physicist and haven't tried to explain this stuff for years, and I can't think of any convincing elementary argument to show that this is the case. Does anyone know of an elementary approach to this? (i.e. one that does not contain the expressions "Lagrangian" and "Noether's Theorem".)
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Suppose someone suggests that following a perfectly elastic collision, two billiard balls are each traveling twice as fast as they were before (and opposite to their original directions). You can't prove him wrong using conservation of momentum, but you can prove him wrong using conservation of energy. Therefore conservation of energy has implications that conservation of momentum does not.
Suppose we have two identical billiard balls, one traveling east and one traveling west, with the same velocities and they collide. Suppose someone claims that both balls will now travel west, at the same velocity. You can't prove him wrong using conservation of energy, but you can prove him wrong using conservation of momentum. Therefore conservation of momentum has implications that conservation of energy does not.
WillO
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Give him an example of inelastic collisions and explain why momentum is conserved but kinetic energy is not. If you explain the reasoning, (all forces are internal hence momentum is conserved) and that there are losses so KE of the system is lost in heat/sound/other forms of energy.. he should get the idea that the two are different beasts. At such a level it is best to illustrate with counterexamples.
yayu
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For a particle in a 1D external time-dependent field there are no energy and momentum conservation laws, yet there are two independent conserved quantities. It is because the differential equation is of the second order and it is accompanied with two independent initial data - the initial position and initial velocity. See an example here.
Vladimir Kalitvianski
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Consider for simplicity a non-relativistic collision between two point particles of same masses in their center-of-mass frame. From total momentum conservation, we know that the center-of-mass (COM) frame is an inertial frame. Moreover, if particle $1$ at some instance $t$ has position ${\bf r}_1$ and velocity ${\bf v}_1$ (relative to the COM frame), then particle $2$ is completely dictated to have opposite position ${\bf r}_2=-{\bf r}_1$ and opposite velocity ${\bf v}_2=-{\bf v}_1$. So from the COM perspective, the two-particle system is completely determined by knowing the state of particle $1$ alone.
Up until now, we have only used momentum conservation, and it doesn't matter whether the collision is elastic, partially elastic, or inelastic. The above is true regardless.
Now let us investigate a collision at initial and final instances $t_i$ and $t_f$ well before and well after the collision takes place. Note that we have already completely extracted all the information in the momentum conservation law to conclude that whatever the particle $1$ does, the particle $2$ would do the opposite. There are no more information available. In particular, momentum conservation gives us no clue about how initial and final velocity of particle $1$ are related.
Finally, let us restrict to an elastic collision. The kinetic energy conservation is in this context the independent statement that the initial and final speed $v_{1i}=v_{1f}$ of particle $1$ are equal (still measured relative to the COM frame).
Qmechanic
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All the following explanation is for elementary students. First ensure that he/she understands that momentum is a vector quantity and energy is scalar. Also s/he might be knowing Newton's 2nd law of motion. Momentum of a system is unique in given direction. For a given system if net external force (on the system) is zero then net momentum of the system does not change. He can argue that so what energy will also not change. Yes, but we can have situation where we apply force on a body and its momentum is changed but energy is not. (A force applied perpendicular to the direction of motion all times). I hope this sets the sail for discussion.
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Momentum and energy are both different depending on what I compare the motion of an object with. If I'm in a train, I have no momentum or kinetic energy relative to the train. Relative to the fields outside the train, however, I have lots of momentum and energy. If I jump off the train, I will come to a stop relative to the fields, so the momentum and energy relative to the fields have to go somewhere. On the other hand, I will then be moving relative to the train, so the momentum and energy to make that happen also have to come from somewhere.
The difference between the momentum and the energy comes from the fact that the forces that change my speed have to act both for a certain amount of time and over a certain distance.
Suppose the force that slows me down is constant, and that the force doesn't make me spin or break me up into pieces. This is the sort of wild idealization that gets Physics a bad name with 16 year olds, but it's a first approximation from which we can go on to a second approximation that's better, and no-one has yet thought up a better first approximation. If the train travels twice as fast, the force has to act for twice as much time to bring me to stop, that's the change of momentum, but the force has to act for four times the distance to bring me to a stop, that's the change of energy.
This gets very tricky, because someone in an airplane that's moving really fast sees the force acting for the same amount of time as someone standing in the field sees the force acting for, but the person in the airplane sees the force acting for much more distance, because when the force started I was right next to the airplane, say, but when the force ended I was a long way behind. So, the change of energy from the point of view of the person in the airplane was much bigger than someone in the fields thinks it was, even though everyone agrees that the change of momentum was the same.
To switch to a different analogy, the energy is important because it determines how far it takes me to stop a car using the brakes, so it determines whether I hit the brick wall that I suddenly see in front of me. The momentum determines how much time it takes to come to a stop, but I can't immediately think of a really graphic situation when that's important.
One can construct different situations endlessly. It can be done in equations, of course, but you'll have to decide whether that's appropriate. I'll be interested if there's anything about this Answer that you think could be made clearer. It certainly isn't complete. Welcome from an Englishman in the USA.
EDIT: Of course overnight I realize that I mention conservation not once. From the point of view of the above it's enough to note that both can be understood to be because of Newton's third law, which is, from Wikipedia, "The mutual forces of action and reaction between two bodies are equal, opposite and collinear". As a result, we can say that the energy added to an object is taken away from the other object, and the same for momentum. The independence of the two conservation laws is essentially just because the two quantities are independent.
I've decided to add a few simplified equations, $$Force = Mass \times Acceleration,$$ $$Kinetic\ Energy = \mathsf{The\ Sum\ Of}\ The\ Forces\ Applied \times The\ Distance\ each\ Force\ is\ Applied\ For,$$ $$Momentum = \mathsf{The\ Sum\ Of}\ The\ Forces\ Applied \times The\ Length\ of\ Time\ each\ Force\ is\ Applied\ For,$$ or, as vector equations, almost certainly beyond what you need, $$\underline{F}=m\underline{a}, \qquad E=\int \underline{F}(t,\underline{s}(t))\cdot\frac{d\underline{s}(t)}{dt}dt, \qquad \underline{P}=\underline{F}(t,\underline{s}(t))dt.$$
Really, this stuff should be left to specialist educators, the best of whom will take time not only to create new ways to explain ideas, but also to study how well different strategies of explanation work for different kinds of student, but I've always been interested occasionally to put myself in this mindset. It's always humbling to discover how much creativity is needed to do it well.
As often, trawling around in Wikipedia, starting from the page on Newton's laws that I mention above, will render up some gems amongst the too-much-information for the purposes of your Question. I particularly like the tail-end comment that "Conservation of energy was discovered nearly two centuries after Newton's lifetime, the long delay occurring because of the difficulty in understanding the role of microscopic and invisible forms of energy such as heat and infra-red light."
Peter Morgan
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Consider elastic collision of a small mass with a large mass (say, fast bullet with (initially) non-moving heavy metall ball). I guess you are considering Newtonian mechanics, so there are following options here:
a) bullet stops after collision, and heavy ball starts moving. If you will require that energy is conserved than you will see that momentum is not conserved, and vise versa. That means that in elastic collision the bullet cannot stop and transfer all its energy (or all its momentum) to the heavy ball because in this case both energy and momentum cannot be conserved.
b) Heavy ball remains not moving after collision, while the bullet is moving with the same velocity in arbitrary direction. Obviously, the energy is conserved, but momentum is not.
Hence, we can imagine the number of outcomes where only one quantity is conserved. But nature leaves us only one choice out of many because both energy and momentum need to be conserved. There are many scenarios where only momentum (or only energy) is conserved, but if we require that both are conserved, only one scenario remains possible.
It is also interesting to consider the situation where the momentum is seemingly not conserved. The simplest case is when a man (or woman) stays on the floor and at some moment of time start moving (walking). The initial momentum of a man (women) is zero, and the final momentum is not. What happens to conservation of momentum? Here it is important to consider a concept of closed system, because only in closed system the momentum is conserved. In this case the closed system includes the Earth. So when we start walking we move the Earth! :-)
Murod Abdukhakimov
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Conservation of energy is almost self-evident. Conservation of linear momentum comes from the fact the in an isolated system of particles there is no preferred location. To try to make a point that the two are related seems that would invoke some really convoluted arguments. One deals with invariance of energy and the other with invariance of location.
John Alexiou
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Check out Noether's Theorem.
Conservation laws are best explained as consequences of space and time symmetries: Because space is the same in every direction we see momentum as beng conserved, because time is uniform we see energy as being conserved. http://en.wikipedia.org/wiki/Noether's_theorem
Note that energy is not conserved in relativistic situations but momentum is - this is related to the nature of relativity.
JonB
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The 16 year old is on to something but no one realizes it because physicists and those who teach physics assume that kinetic energy is a valid scientific principle. This opening sentence will naturally solicite laughter and other reactions but take a moment anyway to consider the following analogy. It looks at what happens when you define energy or force with respect to distance. Energy (kinetic energy) is related to force acting through a distance. Imagine a passenger sitting in the rear seat of a car and he hands his cell phone to the driver. If the car is stationary, that phone travels about 3 feet with respect to the road. If the car is moving, the phone might travel 10, 20, 30 or more feet. The time it takes to hand the phone does not change; it is a constant. Now think about force, it causes things to accelerate. When a body accelerates, it changes its velocity/speed. As this occurs, that body will travel a certain distance and take TIME to do so. First, take wind resistance and other unrelated things out of the mix. Accelerate a body from 0 mph to 10 mph; this takes time and the force will act through a distance. Accelerate the same body from 10 mph to 20 mph using the same amount of force. The time to change that body's speed by 10 mph does not change; the distance it travels during that act will. In short, there is something wrong with the kinetic energy formula. Do not do what everyone does and assume that because something has been around for a long time that it is correct.
