Derivative function of a free falling object colliding with the ground

[NeoPTLD]

Does anyone have a realistic graph of derivative function of falling objects as it falls and strikes the ground in fine resolution?

Let's say one for for a cinder block falling down from 10th floor and disintegrating and another for a basket ball.

In either case d/dx is a linear increase and (d/dx)^2 is 1.00G until the object strikes the ground.

It seems like something like cinder block would come to rest instantaneously but that would give infinity for 2nd and 3rd derivative. In real life, it requires a very small amount of time. Anyone have an idea what realistic (d/dx)^2 and (d/dx)^3 function looks like? I'm guessing significantly steeper than a car crash.

 

[Bobthelost]

It'd be a bitch to model as the block would disintegrate, changing from one uniform block to a host of tiny little ones. You could work out the velocity of the block at impact then work out how long it would take to penetrate the surface and then use that as the time it would take to decelerate. Unscientific, crude and probably wrong, but it's an answer.

 

[dkozloski]

Maybe use a concrete block instead. You could do a study of the actual sizes and densities of the aggragate instead of a general assumption of small particles. Of course it could make a huge difference if the block landed on a corner rather than a flat side as the disintegration of the corner combined with the increasing cross section as the corner collapsed would add a whole host of new variables. I think you could make a career out of this study. In the interest of accuracy you might want to place the block in a ballistic nylon bag to catch all the pieces so the block could be reassembled and the trajectories of the ejected particles could be calculated.

 

[CycloWizard]

It is possible for real systems to have discontinuous derivatives. Indeed, this is most often the case in non-textbook engineering problems. Those with all derivatives smooth and continuous (called "category A" problems) are typically found only in textbooks, not reality. However, I think most of the derivatives in this case should be continuous.

That said, in reality whatever you drop will oscillate when it hits the ground. It will spring back up, then go back down (all very short distances) until friction has had its way and the kinetic energy has been transformed to heat. Thus, consider that it will have a decaying sine waveform for displacement. The velocity will have a decaying cosine form, and the acceleration will have a negative decaying sine (third derivative, of course, will have negative cosine). Anyway, I gotta run now, but maybe I can whip up a sample solution when I get back.

 

[dkozloski]

Why not assume a dead soft, homogenous, copper sphere dropped on the armored deck of a battleship? Now you will be dealing with deformation rather than disintegration. All you have to choose is whether to consider the case of the leading edge or the trailing edge of the sphere.

 

[DrPizza]

I agree with dkozloski,
And, the physics involved would be mind boggling as you went into more and more detail.
About all I remember from undergrad ceramics engineering about impacts is that the physics involved are much more complicated; example: shockwaves.

But for your original question, yes, it does "decelerate" as the ground gives (indents) a little bit, and the cinderblock also deforms slightly during that first very very short amount of time. Then, it's no longer a cinderblock as it begins shattering. At this point, I highly doubt you will be able to advance science very much. The system exhibit an incredible degree of chaos. (Or, if you think not, drop 100 cinderblocks from the same height, onto the same surface. I highly doubt you're going to get any identical results.)

 

[CycloWizard]

IA simple way to examine such a system is to plot the displacement of the object with time by assuming some form of the solution. For example, I took the form x=-sin(wt)*exp(-At). This could represent the trailing edge of a basketball, where x=0 is the location of the trailing edge relative to the ground when the leading edge hits the ground. The value of w is the frequency of the bouncing, while A is a time constant dictating how rapidly the bouncing decays with time. If you plot this function, you'll find that it's a decaying oscillator, which seems appropriate for a bouncing ball when friction effects are included. Try A=3 and w=10 for starters just to get an idea of what it looks like. This form of the solution has the added benefit of easily calculated derivatives.

If you want to get into the dropping of a material and modeling its fracture on impact, things become very complex indeed. Typically when mechanical engineers consider fracture mechanics, it is in the absence of rigid body motion and in an equilibrium sense. In this frame of reference, the second spatial derivative of the displacement goes to infinity at the point of fracture (since the strain is discontinuous at this point). If you were looking at the situation you described, I don't believe the displacement could be described by a continuous function. Rather, it must be described in a piecewise function. The displacement would have a parabolic form (constant second time derivative). During the finite time of deformation and subsequent fracture, I can't say what the displacement function might look like, as it would depend on the point of the block you're considering, as well as the material properties of the block (among other things). Let's just say that this behavior probably isn't linear, and that you could end up with a negative (upward) velocity when all is said and done, though you wouldn't necessarily. After fracture, the piece you're considering would once again start accelerating towards the earth with constant acceleration, assuming it was not interacting with other parts that had broken off.

 

[kevinthenerd]

Break it into finite elements. FEM analysis is easier than the statistical averaging required to model disintegration.

For the basketball, you have a certain amount of deformation in the object and a coefficient of restitution. It's nice if you can find a high-speed CMOS image of that sort of thing. I'd pull down the $40000 camera I have sitting next to me in the lab where I am, but I'm working on studying. (This is just a quick break.)