Change of Momentum in relation to Force

[clm222]

I was thinking about the impact of two bodies. Let's say that we don't know the Coeffient of Elaticity. We know the mass of the object being hit "m1", the the object hitting m1, "m2".
given the velocity of m2, "v", what would the force on m1 be?

First let's assume that m1 is some plane or another, that cannot be moved. (Elasticity=0 "ε")
since ε=0, the object will not bounce off of the surface. Therefore we can assume that the final velocity of m2 will be 0. therefore the (de)acceleration is equal to the incident velocity of m2. Newtons 3rd law states that's the the negative force that deaccelerates m2 is equal to the force m2 is applying to m1. Therefore m2 is applying [m2*v] units of force, is it? would the mass for the force (F=ma) on m1 be the mass of the object hitting m1 or the object itself, m2?

also, for kilograms and meters per second, do you get Newtons in force?

Thanks.

 

[nasu]

therefore the (de)acceleration is equal to the incident velocity of m2.

No, the deceleration is not incident velocity. Acceleration and velocity are different quantities.
Acceleration is the change in velocity over the time interval that change happens.
a=Δv/Δt

You cannot calculate the force from the data you assume known.
This problem was discussed many times on the forum. Look for "impact force" or similar.

 

[supernova1387]

I was thinking about the impact of two bodies. Let's say that we don't know the Coeffient of Elaticity. We know the mass of the object being hit "m1", the the object hitting m1, "m2".
given the velocity of m2, "v", what would the force on m1 be?

First let's assume that m1 is some plane or another, that cannot be moved. (Elasticity=0 "ε")
since ε=0, the object will not bounce off of the surface. Therefore we can assume that the final velocity of m2 will be 0. therefore the (de)acceleration is equal to the incident velocity of m2. Newtons 3rd law states that's the the negative force that deaccelerates m2 is equal to the force m2 is applying to m1. Therefore m2 is applying [m2*v] units of force, is it? would the mass for the force (F=ma) on m1 be the mass of the object hitting m1 or the object itself, m2?

also, for kilograms and meters per second, do you get Newtons in force?

Thanks.

Use principle of linear momentum to avoid confusion. If you want to find the IMPULSE on each body, then we can say:

m2(-Vy)1+∫Fydt=m2(Vy)2

Assuming that the ball is falling vertically so we have impact only in y-directions(upward impulsive force). Since (Vy)2=0, the impulse on the ball would be:

m2(Vy)1=∫Fydt

The same impulse is exerted on the Earth by the ball in the opposite direction. We can't use principle of linear momentum to find this impulse though. Generally, when we have infinity mass like plane, Earth etc, linear momentum doesn't help so we use coefficient of restitution to find the final velocity of the colliding ball etc(if the collision is not plastic e≠0). But for the applied force on the Earth, use what I gave you above to find the force on the ball and then you can conclude the same force in opposite direction is applied on the Earth.

Hope this helps

 

[nasu]

Unfortunately what you gave there cannot be used to find the force on the ball or on the Earth.
You can find the integral of the force over the collision time.
If you know somehow the collision time, you can calculate some sort of average force (divide the change in momentum by this time). If you need to find the actual variation of force versus time, then you need some realistic modeling of the deformations during collision. Some finite element analysis packages can do this.

 

[supernova1387]

A good question you can ask yourself is how finite element package is supposed to calculate the given force variation for you? Of course it would employ some sophisticated numerical methods, but the principle would be the same.

You can calculate the exact force variation if you are given sufficient info with what I said.

here is a simple example. Newton's second law says:
F=m*(dv/dt) → Fdt=mdv (linear momentum principle)

If you know velocity variation in terms of time, you can find force variation too. Let's say v after impact varies according to the following formula:

v=V₀*exp(-a*t), a>0, V₀: pre impact velocity

You can now use simple calculus to find force variation after impact.

 

[nasu]

If you know velocity variation in terms of time, you can find force variation too.

Yes. IF.

The problem is to find the velocity as a function of time. It is part of the same iteration process. You find the force at each step and then how this force changes the velocity and then how the force changes due to the displacement during that step and so on.
At each step you need to take into account the stiffness of the objects (including the non linearity and the plastic limit, if you want to include the non-elastic collisions) as well as the some non-elastic dissipation (even the "most elastic" collision converts some kinetic energy into other forms).

I did not say that a FE program is the only way. You can do it by hand, in principle.
But no matter if you use or not the FE, you cannot solve this kind of problem just by conservation laws or by using the impulse-momentum equation.

Maybe one of us misunderstood the OP. I understand that he assumes he knows the final velocity, after collision. He did not say that he knows the velocity as a function of time, during the collision.

 

[supernova1387]

I'm not sure how FE softwares work exactly , but obviously you need to give it some input and define some constraints for it to work.

I did not say that a FE program is the only way. You can do it by hand, in principle.
But no matter if you use or not the FE, you cannot solve this kind of problem just by conservation laws or by using the impulse-momentum equation.

Again it depends on the problem. I don't think you can solve what OP says by FE either because we have no info. I mean even if we consider other than conservation of linear momentum.

One additional note is that we have to separate particles from rigid( and non-rigid) bodies. The ball in this problem can be thought of as a particle and we can neglect deformation, heat transfer etc. If we can't ignore the size of the ball, the problem would be much more difficult and we then need some FE analysis especially if the ball is non-rigid. Then you need Young's modulus etc.

And btw, if the ball is brought to rest after impact and you want to calculate deformations etc, that problem won't be dynamic anymore. It will be static and you need to use mechanics of materials.

 

[nasu]

I'm not sure how FE softwares work exactly , but obviously you need to give it some input and define some constraints for it to work.

Of course. Did I imply otherwise?

Again it depends on the problem. I don't think you can solve what OP says by FE either because we have no info. I mean even if we consider other than conservation of linear momentum.

He did not give specific data but hinted that you know about the elasticity of the bodies.
He said bodies and not "points".

One additional note is that we have to separate particles from rigid( and non-rigid) bodies. The ball in this problem can be thought of as a particle and we can neglect deformation, heat transfer etc.

So you will have the collision of two "particles"? I am not sure what do you mean by this How do you imagine a collision in this case? Do you mean elementary particles?

If we can't ignore the size of the ball, the problem would be much more difficult and we then need some FE analysis especially if the ball is non-rigid. Then you need Young's modulus etc.

But this is the case to consider for the problem proposed by the OP. If you want to find the force exerted during the collision, this is what do you need to consider.
Do you have a model for collision of pointless particles?

And btw, if the ball is brought to rest after impact and you want to calculate deformations etc, that problem won't be dynamic anymore. It will be static and you need to use mechanics of materials.

Well, the final deformation is the results of the collision, a dynamic process. You don't have a static, known load applied to the object. So you will have the same problem as for the other cases.
And I thought you will be using "mechanics of materials" for both dynamic and static case.

PS.Maybe it will be simpler if you describe the actual problem you have in mind. We may be talking about different things. The part about collision of particles, that you mentioned, is confusing.

 

[supernova1387]

He did not give specific data but hinted that you know about the elasticity of the bodies.
He said bodies and not "points".

Bodies can both refer to particles and rigid/non-rigid materials. What he meant by coefficient of elasticity ε was actually coefficient of restitution and particles can have ε too. I don't think he meant Young's Modulus(elastic modulus) as that doesn't make sense for this kind of problem. He says the body comes into rest after impact and assumes ε=0 which means we have a plastic impact and that has nothing to do with Young's Modulus etc if you are taking elasticity in that regard.

So you will have the collision of two "particles"? I am not sure what do you mean by this How do you imagine a collision in this case? Do you mean elementary particles?

What I mean by particles is quite clear. We use it many times in dynamics and in many textbooks we first talk about dynamics of particles and then go into rigid bodies . But anyway, it's an idealized body that occupy a single point. Let me ask you a question. How do you calculate gravitational force between the Earth and the Mars? Do you consider them as rigid bodies or do you assume all their mass is concentrated at a single point and then find the force? Are they elementary particles here? What about a car going around a circular track? What about a a boy sliding down a slide?

We can assume a body as a particle if it has a negligible size or the size of one is negligible compare with the other etc or in the planetary example above if their distance is much bigger than the size of the bodies. In this problem, we have a ball and surface of the Earth. The ball has negligible size compare with the Earth, so you can consider it as a particle. Unless you want to examine the internal deformation within the ball, there is no need to take it as a rigid body. And you can't analyze the internal deformation until you specify force variation. You can find the impulse on the ball accurately with conservation of linear momentum , but to find force variation, we need to have some more info like velocity variation etc. That may need some experiments to find something like the velocity equation I gave earlier or maybe your FE software can estimate that for you.

Well, the final deformation is the results of the collision, a dynamic process. You don't have a static, known load applied to the object. So you will have the same problem as for the other cases.

Because the body comes into rest after impact, the problem would be static. In this case, you have a non-linear applied force to the body ( and by body here I don't mean particles). This is how I would model it. But you first need to find this non-linear force and without that you can't find the deformation etc.

nd I thought you will be using "mechanics of materials" for both dynamic and static case.

Not if your body is a particle. In classical mechanics we assume a particle has no internal structure.

PS.Maybe it will be simpler if you describe the actual problem you have in mind. We may be talking about different things. The part about collision of particles, that you mentioned, is confusing.

Have a look at the examples I gave above. That will hopefully make it clear by what I mean about particles.

 

[nasu]

What I mean by particles is quite clear. We use it many times in dynamics and in many textbooks we first talk about dynamics of particles and then go into rigid bodies . But anyway, it's an idealized body that occupy a single point. Let me ask you a question. How do you calculate gravitational force between the Earth and the Mars? Do you consider them as rigid bodies or do you assume all their mass is concentrated at a single point and then find the force? Are they elementary particles here? What about a car going around a circular track? What about a a boy sliding down a slide?

We can assume a body as a particle if it has a negligible size or the size of one is negligible compare with the other etc or in the planetary example above if their distance is much bigger than the size of the bodies. In this problem, we have a ball and surface of the Earth. The ball has negligible size compare with the Earth, so you can consider it as a particle. Unless you want to examine the internal deformation within the ball, there is no need to take it as a rigid body. And you can't analyze the internal deformation until you specify force variation. You can find the impulse on the ball accurately with conservation of linear momentum , but to find force variation, we need to have some more info like velocity variation etc. That may need some experiments to find something like the velocity equation I gave earlier or maybe your FE software can estimate that for you.

OK, now it's clear what you mean.
However, for collisions this model, of point-like is quite useless.
You can treat collisions in two ways:
1. As a black box where you input initial velocities and masses and output final velocities.
The only think you need to know about the box is the restitution coefficient.
Conservation laws will provide the final velocities.
For this treatment you can consider the bodies points or anything you want. It does not matter anyway. Conceptually, it does not make much sense to have restitution coefficient for point-like particles but again, it does not matter for the math and the coefficient is usually measured experimentally.

2. Consider the deformations of the bodies during the collision and the time variation of the velocities from the initial values to the final.
To do this specific information about the material properties are necessary.

Treatment 1 does permit to answer questions about the impact force unless you measure some extra parameters experimentally. And then the most straightforward thing may be to measure the actual force and forget about any theoretical calculations. Alternatively, measuring the speed variation can provide the force, as you suggested.

Treatment 2 allows to find, at least in principle, the details about the impact force.

The question is if the OP meant to find the impact force by doing measurements of time variable parameters or by just knowing the final values of the velocities. I read the post to mean the last case.