Re http://hyperphysics.phy-astr.gsu.edu/hbase/balpen.html

I was helping my daughter solve a ballistic pendulum problem last night.

I'd never seen one of these before, so I saw it as an opportunity to teach her the value of starting from first principles.

In my naivete, I approached the problem first from conservation of energy. I got the wrong answer.

My second attempt used conservation of momentum, and the answer I arrived at was correct.

I am trying to come up with a simple explanation of why CofE doesn't work, and a rule or rules to use (say during an AP exam) as to whether to use CofE or CofM for a particular problem, and a method to determine that the correct choice was made.

Any suggestions? @WBahn, maybe?

High school level, BTW.

 

Further, am I correct in assuming that in elastic collisions both CofE and CofM are applicable, and in non-elastic collisions only CofM?

 

Correct - the simple rule is of course that the (in)elasticity of the collision is the determinant, but the more I think about it the less that explains. I always thought I understood it, but now I see I merely accepted it. Work needed......

 

Correct - the simple rule is of course that the (in)elasticity of the collision is the determinant, but the more I think about it the less that explains. I always thought I understood it, but now I see I merely accepted it. Work needed......

If so, that implies the energy absorbed and/or radiated away (heat, sound, etc.) in any inelastic collision is always determinate regardless of the nature of the bodies or their composition, yes?

 

And, therefore: if CofM gives a different answer than CofE, does that imply an inelastic collision, and that CofM is the correct method?

 

Finally, if a problem can be solved using CofM only, is this always sufficient even if a solution could also be had using CofE?

In other words, try CofM first, and if no solution can be found, use (or incorporate) CofE?

 

Last question: what is special about CofM that it accurately ignores energy losses in inelastic collisions?

 

...the energy absorbed and/or radiated away (heat, sound, etc.) in any inelastic collision is always determinate...

Determinate as in being the difference in Ek before and after.

what is special about CofM that it accurately ignores energy losses in inelastic collisions?

That's the rub - I get Noether covers it mathematically, but it stills lack 'Feynman' clarity.

 

Here's a start:

https://profoundphysics.com/why-is-momentum-conserved-but-kinetic-energy-is-not/

 

Re http://hyperphysics.phy-astr.gsu.edu/hbase/balpen.html

I was helping my daughter solve a ballistic pendulum problem last night.

I'd never seen one of these before, so I saw it as an opportunity to teach her the value of starting from first principles.

In my naivete, I approached the problem first from conservation of energy. I got the wrong answer.

My second attempt used conservation of momentum, and the answer I arrived at was correct.

I am trying to come up with a simple explanation of why CofE doesn't work, and a rule or rules to use (say during an AP exam) as to whether to use CofE or CofM for a particular problem, and a method to determine that the correct choice was made.

Any suggestions? @WBahn, maybe?

High school level, BTW.

The problem with the use of conservation of energy is that it doesn't take into account all of the relevant kinds of energy. It assumes that kinetic energy is converted completely to gravitational potential energy. But what about energy that is converted to heat?

Imagine two identical objects traveling toward each other at the same speed that collide and stick together (think to lumps of soft clay). The objects, ideally, simply stop where they are. All of their kinetic energy has been converted to heat via deformation. But conservation of momentum always holds.

 

Last question: what is special about CofM that it accurately ignores energy losses in inelastic collisions?

It is a direct consequence of Newton's Third Law of Motion. Because two objects that interact always exert equal and opposite forces on each other, that means that the impulse delivered to one (impulse being the integral of force over time) is always exactly opposite the impulse delivered to the other (since impulse is a vector quantity). An impulse applied to an object is equal to the change in that object's momentum, hence the change in momentum of one object is always equal and opposite to the change in momentum of the other object, and therefore the total change of momentum of the system is zero.

The total change of the energy in the system is also zero, but it must be summed over all possible forms of energy, and identifying which forms of energy are impacted can be tricky.

EDIT: Fix typo.

 

The problem with the use of conservation of energy is that it doesn't take into account all of the relevant kinds of energy. It assumes that kinetic energy is converted completely to gravitational potential energy. But what about energy that is converted to heat?

So, why do we know, in the case of the Ballistic Pendulum, that not all the KE of the projectile [1/2(m1)v1^2] was first transferred to the pendulum (+ projectile) [1/2(m1+m2)v2^2] and then converted to GPE [(m1+m2)gh]? Is this just because they stuck together?

I'm curious if this is analog to the case of charging one capacitor with another, the process of which always results in a loss of energy, even though charge stays the same.

 

It is a direct consequence of Newton's Third Law of Motion. Because two objects that interact always exert equal and opposite forces on each other, That means that the impulse delivered to one (impulse being the integral of force over time) is always exactly opposite the impulse delivered to the other (since impulse is a vector quantity). An impulse applied to an object is equal to the change in that object's momentum, hence the change in momentum of one object is always equal and opposite to the change in momentum of the other object, and therefore the total change of momentum of the system is zero.

The total change of the energy in the system is also zero, but it must be summed over all possible forms of energy, and identifying which forms of energy are impacted can be tricky.

Yeah. I think I remember this now from high school physics over 100 years ago. Stuff like this gets stale when you don't use it in practice often.

 

Is my assertion in Post #6 correct?

 

So, why do we know, in the case of the Ballistic Pendulum, that not all the KE of the projectile [1/2(m1)v1^2] was first transferred to the pendulum (+ projectile) [1/2(m1+m2)v2^2] and then converted to GPE [(m1+m2)gh]? Is this just because they stuck together?

It's a pretty safe bet that anytime objects stick together that the collision was inelastic. But this is also a bit of cart-before-the-horse.

A collision in which kinetic energy is conserved is, by definition, an elastic collision. Thus any collision in which kinetic energy is not conserved is an inelastic collision.

The reason that all of the KE of the projectile can't be transferred to the pendulum, under the constraint that the bullet embeds into the block, is that there is no velocity at which the combined block can move that satisfies the conservation of momentum.

I'm curious if this is analog to the case of charging one capacitor with another, the process of which always results in a loss of energy.

Essentially. But it's best to think of it not as a loss of energy, but as a transformation of energy to some other form that hasn't been properly accounted for.

 

But it's best to think of it not as a loss of energy, but as a transformation of energy to some other form that hasn't been properly accounted for.

Of course. That was just lack of precision in my words.

 

Finally, if a problem can be solved using CofM only, is this always sufficient even if a solution could also be had using CofE?

In other words, try CofM first, and if no solution can be found, use (or incorporate) CofE?

CofM is essentially the gold standard, because it's much harder to miss things that needed to be taken into account. When using CofE arguments, this is always a concern because there are so many different forms of energy that could be at play.

But, when CofE arguments are valid, it is often considerably easier to arrive at the answer than using CofM because, if nothing else, energy is a scalar quantity while momentum is a vector quantity.

 

to whether to use CofE or CofM for a particular problem, and a method to determine

I have a simple rule in mind: in nature nothing is lost, nothing is gained, everything is transformed.

To break a glass, or to deform a car, I need energy.
So in inelastic collisions part of the kinetic energy is transformed into deformation energy.
But the mechanical momentum is conserved because the impulse represents the "quantity of movement" m*v

Examples?
With what force does a 1 kg and 0.5 m long bird hit an airplane flying at 720 km/h?
Many believe that it is not verybig.
What if I told you that it hits with tons force?

 

Elastic collisions. I pause this video and ask:

If I lift a ball from one end and let it fall, how many will rise at the other end?
One!
Why?
I get the wrong answer that he has no one to hit anymore.

But if I lift three balls from one end and let them fall, how many will rise from the other end?
Try to guess before you see what happens in the video.

 

Elastic collisions. I pause this video and ask:

If I lift a ball from one end and let it fall, how many will rise at the other end?
One!
Why?
I get the wrong answer that he has no one to hit anymore.

But if I lift three balls from one end and let them fall, how many will rise from the other end?
Try to guess before you see what happens in the video.

I am well aware of the physics of Newtons Cradle, and I know how to compute the only solution such that both momentum and kinetic energy are always preserved.

This is a substantially different case than the ballistic pendulum, but I didn't immediately recognize the differences when presented with my daughter's homework problem.

As I said in a previous post, it's been a really, really long time since high school physics.