Paradox with Newton's E=1/2mv2 ? Looking for clarification

1. Jun 27, 2010

lenfromkits

paradox with Newton's E=1/2mv2 ??? Looking for clarification...

I don't understand why these points of view don't balance. This is non-relativistic.

If two objects are travelling towards each other from opposite walls of a room, then the sum of their Kinetic energies relative to the room does not seem to match the kinetic energy relative to each other.

For example. relative to the room, if each has a mass of 1Kg and is travelling at 10m/s, then since E=1/2mv2, each will have a Kinetic energy of 50J, for a total of 100J.

But relative to each other, they are travelling at each other at 20m/s. Relative either one of the objects, the other then has a Kinetic energy of 200J.

Also note, each object could have acclerated from each wall according to this formula in such a way that it is expected that they each have 50J of energy. But according to the formula, if the two collide (and stop), they ought to release 200J of energy into another form.

???

Thanks.

Last edited: Jun 27, 2010
2. Jun 27, 2010

cmos

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

I would like somebody to confirm my reasoning here, but I believe that the answer is that, in Newtonian mechanics, space (and time) are absolute. In other words, in solving a mechanics problem, you must make sure you analyze the "proper" system. The proper system, in your example, being the one in which the walls of the room aren't moving (i.e. the one which gives E = 100 J).

Now, you may say, "Ah, but the walls are only for convenience! Let's take away the walls and place our particles in deep space; then what?" Well, then to properly analyze the system, we would still have to find some other object to act as a "stationary wall;" perhaps the nearest star or the cosmic microwave background radiation.

Of course, this is all non-relativistic. Relativity takes away these inconsistencies (or perhaps, rather, inconveniences) by toppling the notions of absolute space and time for a relative spacetime.

3. Jun 28, 2010

lenfromkits

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

Wow. Great response! Thanks :)

You've confirmed that it is probably is a bit of a conundrum and not simply me being totally out to lunch. I wonder if Relativity really solves that though given its paradoxes. They seem to kind of stem from the same neighborhood and maybe Einstein just traded one problem for another.

4. Jun 28, 2010

Staff: Mentor

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

This is true in both non-relativistic and relativistic mechanics. Why do you think this is a problem?

5. Jun 28, 2010

bp_psy

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

This is only non-relativistic in the sense of special relativity effects.It is relative in the sense of Galilean relativity. You are dealing with two different inertial frames of reference the laws of classical physics are the same in both of them.When you are changing your point of view from the objects moving with respect to a chosen point to the objects moving with respect to each other. You have changed your frame of reference. You will get two different results for both the momentum and KE these two results are both valid depending on which frame you are considering.In conclusion you don't need to resort to anything other then old Newtonian mechanics for this problem.

6. Jun 28, 2010

lenfromkits

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

Thanks bp_psy for your response. This almost makes sense to me except that the Energy contained within one frame of reference could be released (via collision) into the other frame of reference, merging the two views into one and forcing the question to be answered, "how much energy is there?"

Last edited: Jun 28, 2010
7. Jun 28, 2010

lenfromkits

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

Hi jtbell. I'm not sure "Why do you think this is a problem" is really an appropriate answer. Noticing that numbers don't balance in a straightforward way is always up for questioning. Please only respond if you have something constructive to say.

8. Jun 28, 2010

bp_psy

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

The amount of mechanical energy is frame dependent. If I am in a car and I put my hand trough the window my hand has 0 KE with respect to the car. If my hand hits somebody on the sidewalk it will do some work on him. This does not violate any laws of physics though. In your initial collision problem what you will find is that in both of the frames you use you will find that the momentum and the total energy is conserved if the collision is elastic.Which is something really important and in no way paradoxical.

9. Jun 28, 2010

lenfromkits

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

Hi bp_psy. :)... It might not violate laws of physics but I suspect the police might have a different opinion on that! Great analogy.

Okay, it seems this is very clear in your mind and you are surely correct, but I'm not quite seeing it yet. When the two objects collide in my example, they change into the 'room' frame of reference. What happens if there was a pot of water somehow involved in the collision and the kinetic energy was all transformed into heat. How hot would the water be? I would think that both frames would have to agree on that. But, especially since after the collision it is all one frame, they certainly must agree.

If you were to conclude that this is similar in nature to the twin paradox where it has, let's call it "weirdness" then that agrees enough with my understanding and I'm happy. What I don't understand is how this is normal in a classical sense where physics doesn't tend to defy reason like [Einstein] relativity does.

Thanks.

Last edited: Jun 28, 2010
10. Jun 28, 2010

lenfromkits

11. Jun 28, 2010

bp_psy

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

This is not true. A inertial frame of reference is an abstract physical entity that you create and is not changed by the collision. If you pick IFR moving in the same direction and with the same velocity as one of the object this IFR will not be affected by the collision. This means that you can let the IFR continue on its initial direction and do the math in it. You will find that momentum and KE is conserved if the collision is perfectly elastic.
In such a problem you are not dealing with inertial frames of reference. Nothing we talked until now applies.

Well on this I can only say that Einstein relativity doesn't defy reason it is a mathematically rigorous theory supported by a lot of empirical evidence. If you think it defies reason that is your problem.

12. Jun 28, 2010

lenfromkits

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

Hi bp_psy. Sorry, by 'defying reason' I meant that it defies normal senses - hence being the revelation that it was, as opposed to Newton's physics which are a whole lot more intuitive. Time dilation just isn't an intuitive phenomenon. So for me this whole thread doesn't fit into my sense of intuitive but seems more like issues dealt with in Einstein's Relativity.

I don't know what you mean by "nothing we talked until now applies." I'm focusing on the various kinetic energies contained within objects. This energy is supposed to be transferable to other forms via friction, etc. So on one hand to say the objects have 100J of energy and on the other to say they have 200J seems to be a discrepancy since we can take the 'relative' energies and turn them into an 'absolute' energy by converting them into heat.

In other words, relative to one of the flying objects, when the other hits it, the heat from the impact could heat the water by 200J. But relative to the room, there was only enough kinetic energy in the objects to heat it by 100J - especially since the tools in the room that accelerated those objects only applied 50J to each to get them up to speed.

Hmm. By 'change to room frame of reference' I meant that if the two objects collide head on in such a perfect way that all their energy is transformed into heat and the both come to a stop relative to the room - then they would at that point be back in the same frame of reference as the room (same speed, same direction, same place, etc)

13. Jun 28, 2010

bp_psy

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

This is the last post this night so I keep is short.
Consider 2 object of mass m and opposite velocities v.The collision is perfectly inelastic.
In a stationary frame (the room) we have the following equation. (m*v^2)/2+(m*v^2)/2=(m*v^2)=heat after the collision. In a frame that moves with one of the objects we have
KE initial=(m*(2v)^2)/2=2m*v^2. The moving frame is not affected by the collision it continues moving with velocity v. The two object that are stationary with respect to the stationary frame are not stationary in the moving frame they have velocity v. Now that means that the the new combined object mass=2m. The KE after collision in moving frame =(2m*v^2)/2 =m*v^2. Subtracting this from the initial KE we get heat dissipated in the collision=m*v^2. In conclusion no paradox.

Last edited: Jun 28, 2010
14. Jun 28, 2010

Gerenuk

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

I haven't read all the answer, but he is a short one:
There is no relativity needed. Indeed the difference in kinetic energy is different for both frames. But it doesn't matter at all. You might say if the objects collide you can see the energy difference as some kind of deformation in the objects. But the deformation energy is the same. Even though in one frame the relative kinetic energy is higher, but in that frame the lumped objects (after the inelastic collision) keep on moving together so not all of the kinetic energy is going into deformation. On the other hand in the frame where the compound object is stationary after the collision the relative kinetic energy is smallest beforehand, but then there is no energy "wasted" on kinetic energy after the collision.

Last edited: Jun 28, 2010
15. Jun 28, 2010

Staff: Mentor

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

It sounds like you're thinking of conservation of energy. That applies only so long as you stick to a single inertial frame of reference.

Any process can be analyzed in any inertial reference frame, so long as you stick with the same frame from beginning to end. You may be thinking of reference frames as necessarily "attached" to objects, but this is not the case.

If you need to make a reference frame more concrete, think in terms of an observer that is separate from the objects in the collision, who is riding past the collision site on a cart that is moving at constant velocity. This velocity may match the velocity of one of the balls, before or after the collision, but it doesn't have to. If the observer is moving at a velocity which matches ball #1 before the collision (for example), then before the collsion ball #1 is stationary in his frame. During the collision, the ball changes velocity but the observer and his frame "keep going."

In the case of the two colliding balls, you can analyze it from e.g.

--the reference frame in which ball #1 is initially stationary and #2 is moving
--the reference frame in which ball #2 is initially stationary and #1 is moving
--the reference frame in which the center of mass of the two balls is stationary
--or any other inertial reference frame.

The energy of each ball is different in each frame, and the total energy of the system (kinetic plus whatever other kinds of energy are involved) is different in each frame. Nevertheless, in any frame, the total energy remains constant as time passes; but once you've chosen a frame, you have to stick with it in order to make this statement.

Last edited: Jun 28, 2010
16. Jun 28, 2010

Staff: Mentor

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

Hi lenfromkits,

You are already getting good help and information from bp_psy and jtbell, so I don't want to distract from that. But I thought I would let you know the correct words for what you are describing, it may help to know the right words if you decide to consult Dr. Google on the subject.

Frame-variant: describes a quantity whose value changes depending on the coordinate system (aka reference frame) used to analyze the situation. Opposites are "invariant" or "frame-invariant". Energy is a frame-variant quantity.

Conserved: describes a quantity whose value does not change over time or across some interaction like a collision. Energy is a conserved quantity.

PS Objects do not enter or leave a reference frame. A reference frame is a coordinate system, so it goes on forever. An object is "in" all reference frames at all times simultaneously. It may have different positions or velocities in different reference frames, but there is no sense in which an object is not "in" any of them.

17. Jun 28, 2010

my_wan

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

But wait, if they are traveling at each other at 10m/s relative to the room, and 20m/s relative to each other, this is wrong.

Room frame (Galilean):
Person1 10m/s = 50J
Person2 10m/s = 50J
Total 100J

Person1 frame (Galilean):
Person1 0m/s = 0J
Person2 20m/s = 100J
total 100J

Person2 frame (Galilean):
Person1 20m/s = 100J
Person2 0m/s = 0J
total 100J

18. Jun 28, 2010

Staff: Mentor

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

ok

½*(1)*(20)² = 200 J

19. Jun 28, 2010

bp_psy

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

I had actually made an error there.(Why I was posting here at 2 am?:surprised).All the rules of reference frames still apply in this case.If you take into account the dissipative forces in both the reference frames you will find out that the same amount of work was done by them.For example If we shoot a bullet of mass m into a large piece of wood with speed v the work done by friction is (m*v^2)/2 in a frame stationary with respect to the piece of wood. If you take your frame to be moving with speed v in the initial direction of the bullet you will find out that the energy went from 0 to (m*v^2)/2 during the collision.The amount of work done by friction is therefore the same. As others have said this will be true for any IFR. I leave it to you to check the result for a frame that moves in the opposite direction to the bullet and a frame that moves in an arbitrary direction.

20. Jun 28, 2010

my_wan

Re: paradox with Newton's E=1/2mv2 ??? Looking for clarification...

I just copied the numbers off the OP. The point is that in all Galilean cases the total velocity defining the energy of the collision is 20m/s. You add the relative velocities to calc the total energy.