Inelastic Collisions - variables and equations

[resurgance2001]

Ok - this is a moderately tough question which I can't figure out.

So I am trying to work on a simplified model to start with.

I imagine a solid, very massive impenetrable object.

I have a tube or any long object which can exhibit some elastic behavior and also plastic behavior.

If the tube hits the much larger massive object at slow speed I reason that it pretty much just stops dead in it's track - tiny bit of bounce but a low low restitution constant.

Above a certain speed, when the tube hit the wall, the force required to cause plastic deformation is exceeded and so the tube is crushed. The high the speed the more it gets crushed.

Now, finally getting to the question:

If I measure the deceleration of the tube, at the end of the tube, furthest from the wall, so the end that is not being crushed, what equation can I use to calculate the deceleration?

It seems like it is at least a first or maybe second order differential equation because the tube's length is not constant and hence its center of gravity is shifting towards the rear.

Also there must be a point where it slows down to a point where it no longer has enough momentum to cause plastic deformation. But I am just not quite sure where to start.

Cheers and thanks for reading.

 

[resurgance2001]

In order to understand this better, I have watched a few videos about car crashes. It appears that when a car crashes into a solid wall, during the part of the collision where the crumple zone is crumpling, the end of the car does not actually show any deceleration.
However I don't know if this is really true because I am only making an qualitative estimate.

 

[jbriggs444]

The back end of a car that is crashing into a brick wall will decelerate during the collision. It has to. Otherwise it would not wind up at rest.

But the compressing force at the front of the car in such a collision is greater than the compressing force force at the rear of the car. As you can see from your high speed videos, the rear part of the car moves as a rigid unit during the collision. The compressing force at the front of the car is responsible for the deceleration of the entire car. The compressing force at the midpoint is responsible for the deceleration of only the part of the car rearward of the midpoint. The compressing force at the tail-lights is responsible for deceleration of just the tip of the rear tail lights. F = ma. The less mass you have to decelerate at a given rate, the less force it takes to do it.

The car is going to tend to crumple where the compressing force is highest. It is going to remain rigid where the compressing force is not so high.

In addition to this factor, man-made structures often resist compression best when they are intact. Once they've been bent a bit out of shape they resist less well. Welds are broken. Support columns are no longer straight. So if the front is where the car begins crumpling, the front is where the car will tend to continue to crumple.

 

[resurgance2001]

Thanks Jbriggs.

That helps a lot.

I saw another two videos which were cause for thought. There is a video of an F14 crashing into a slab of concrete. It's strange because the front end of the plane is basically atomized as it hits the wall. However, observing the tail section of the plane it really does not show any deceleration.

It reminded me of that famous video of a plane going into the south tower of the WTC. That tail of that plane also seems to show no deceleration.

Notwithstanding, what you said about the force being greater at the end at which the impact occurs is very helpful.

I want to try to set up some really simple labs so I can actually see what happens when one uses real materials. Thanks Peter

 

[PJC01]

I would first commend you for considering a simplified model to start with. This is an important step in scientific inquiry as it allows you to understand the basic principles before moving on to more complex scenarios.

In terms of inelastic collisions, there are several variables and equations that are important to consider. First, we have the masses of the objects involved, which can be denoted as m1 and m2. These masses will determine the amount of momentum each object has before and after the collision.

Next, we have the velocities of the objects, which can be denoted as v1 and v2. These velocities will also play a role in determining the momentum of each object.

One important equation to consider in inelastic collisions is the conservation of momentum, which states that the total momentum of a system before and after a collision remains constant. This can be expressed as:

m1v1 + m2v2 = m1v1' + m2v2'

Where v1' and v2' represent the velocities of the objects after the collision.

In addition to conservation of momentum, we also have the concept of kinetic energy, which is the energy an object possesses due to its motion. In inelastic collisions, kinetic energy is not conserved and is instead converted into other forms of energy, such as heat or sound. This can be expressed as:

1/2m1v1^2 + 1/2m2v2^2 = 1/2m1v1'^2 + 1/2m2v2'^2

In your scenario, where the tube is hitting a solid, impenetrable object, it seems that the collision would be considered completely inelastic. This means that the objects stick together after the collision and move with a common final velocity. In this case, the equations above can be simplified to:

m1v1 + m2v2 = (m1 + m2)v'

1/2m1v1^2 + 1/2m2v2^2 = (m1 + m2)v'^2

Where v' is the common final velocity of the objects after the collision.

Now, to address your specific question about calculating the deceleration of the tube, we can use the equation for average acceleration, which is:

a = (vf - vi)/t

Where vf is the final velocity, vi is the initial velocity, and t is the