Momentum, impact force and Earth

[Jpcgandre]

Suppose two objects, A and B, with large lengths LA and LB, and masses MA and MB, collide at time t0.

Both objects before collision are vertical and aligned concentrically, being object B positioned initially at a higher z coordinate than object A.

The bottom end of object A is rigidly constrained to Earth.

Object B drops with zero initial speed towards object A.

I assumed an ideal inelastic collision between these two objects.

I'm trying to obtain estimates of the collision force, F, at an arbitrary time, ti, after the collision.

The potential issue I'm having is that the value of the collision force just after impact between the closest parts of both objects increases to a large value, much higher than the static value, after which it reduces gradually until the impact wave reaches Earth.

When the latter occurs, both objects experience a large deceleration and thus a large change of momentum and therefore I obtain a large collision force value at this time interval after which it rapidly decreases and tends to the static value.

My question is not about concrete values, just on the principles. Is it to be expected that the collision force increases due to the deceleration caused by the restraint introduced by Earth? In my calculation of the collision force, I just enter with the change of momentum of object B during a time period and the gravity of the mass of object B (dynamic + static parts). Do I need to account for the restraint introduced by Earth in another way, neglect it, or...? How to do this conceptually?
Thank you

 

[Baluncore]

Welcome to PF.

Potential energy will be converted into kinetic energy prior to the impact. The kinetic energy will compress the materials in an elastic way. The energy will be divided along two paths, part at the speed of sound downwards, part at the speed of sound, reflected back upwards.

The critical thing will be the contact time during the collision.

If the yield point of the material is exceeded by the force, the materials will bend and collapse, or melt and flow, possibly becoming a single object, or a wide splatter.

So we need to know the velocity of the objects and the speed of sound in the material before proceeding further with the analysis. Yield point, melting point and thermal capacity may become important during the impact.

 

[Jpcgandre]

Hello Baluncore thank you for the reply.
At this point I'm after the basic example possible. Please consider all materials are infinitely elastic and have a very large yield point so that all material nonlinearities are out of the question.
My question goes to the core of if it is possible that during collision of these two objects the impact force is somehow influenced by the deceleration caused by the restraint introduced by Earth at the bottom and of object A. The impact duration will greatly exceed the time that the sound wave reaches Earth.
If the answer is yes can you please elaborate how to take Earth into account to determine the impact force?
Thanks

 

[Baluncore]

The bottom end of object A is rigidly constrained to Earth.

You do not say what that means about the character of the Earth. Is the Earth assumed perfectly rigid, a thousand times more rigid than diamond ?

Take a look at Newtons cradle; https://en.wikipedia.org/wiki/Newton's_cradle
That is what you are doing with two rods and the Earth. You have two rods that are transmission lines for the pressure waves that will be radiated from the point of collision.

The value of Young's modulus will be important in determining the elastic properties of the objects and how long the collision will last.
https://en.wikipedia.org/wiki/Young's_modulus
Think of material elasticity as being a long slow soft spring, or a short fast hard spring. The contact times will be very different for the same reflected energy.

Don't underestimate the importance of the speed of sound in the material. Just hammering a fence post into the ground involves transmission lines and reflection coefficients.

Until you put some numbers on the dimensions and materials it will not be possible to analyse the situation further.

 

[Jpcgandre]

The ground is assumed to be rigid in comparison with the stiffness of the rods.
I know that impact force depends on the duration of impact and this, in turn, depends on the mechanical characteristics of materials. This is all fine and I'm not aiming to solve analytically or conceptually all possible cases.
My thought example is aimed at trying to understand if it is reasonable to postulate that the impact force between two objects will be influenced by one of them being attached to the ground.
So:
Just after impact, the impact force depends only on the mass of the objects, velocity of the objects and what is the volume of the objects affected by the impact wave. Of course, material and structural resistance play a role (a force only increases as much as resistance is available), but here for sake of simplification, I'm not considering them.
After the impact wave reaches the top end of the top object the velocities of this object will also depend on the structural mechanics of the impact: the change in velocity is faster for more rigid impacts and longer for less rigid impacts. And as a result, different impact forces will be obtained.
After the impact wave reaches Earth (bottom end of the bottom object), the question I have is if the deceleration caused by the restraint introduced by Earth will change (more or less is not my point, just a binary yes or no) the impact force (assuming the impact is still occurring, i.e. a static equilibrium has not yet been met). And if yes, my follow up question is for hints on how this can be considered to determine the impact force: by using the same procedure of change in momentum = net impulse where the net impulse is given by the impact force minus self-weight? Or do I need to add an extra term to the net force? Or something else?
Thanks

 

[Baluncore]

Your question is still too general to answer.
The safe bet would be to say yes, the Earth has an effect, but then it stalls again not knowing the length and speed of sound in the rod. Are the rods the same material and length ?

A pressure wave from the collision will be reflected from Earth back up to the point of the collision. At the same time the impact wave that traveled up the upper rod, will be reflected from the open end, back down to the collision site. Those two reflections will meet, but they will probably have opposed phase as one end is constrained by Earth, while the other is in free space.

 

[Jpcgandre]

Great, that is also my opinion! :)
Rods can be of same material and length.
About the pressure waves, yes reflection occurs but I'm not sure how will this wave influence the impact force (assuming impact is still occurring). How to conceptually account for it in the kinematics equations?
Thanks a lot!

 

[Baluncore]

How to conceptually account for it in the kinematics equations?

I would build an analog model in SPICE from two transmission lines;
Terminations and Zo would be acoustic impedances;
Lines would be joined at the collision point by a voltage source, that steps from initial 1.0 V to zero, at t=0;
One with far end "grounded", the other rod with termination open;
Analyse for the transient response current of the voltage source.

 

[Jpcgandre]

Sorry now I'm totally lost. I'm not electrical engineer... what would be in this example the equivalent to impact force?

 

[Baluncore]

... what would be in this example the equivalent to impact force?

The impact force was the voltage step amplitude at time zero.
I have not thought all the way through my analog model yet.

I wrote a quick test for two identical rods and found that the two rods became one at the instant of impact. If they were not separated before the two reflections returned to the point of impact, then there was no impedance mismatch there, so both waves pass through and continue to the far ends of the combined rod. That one rod has twice the original length, and contains two separate waves separated by 180°, traveling independently in different directions.

It is clear that both reality and the model details are going to need more thought.