I was wondering: If you let a small stone drop on a body of water, record it on film, and replay the scene in slow motion, will it be possible to see the difference with a huge rock that falls, in real-time, in a body of water?
Let's take for the stone a spherical mass, and let's assume that the body of water has a flat surface. The stone hits the water perpendicularly. And let's assume that the rock is a spherical mass too with the same density as the small mass. Classical physics (mechanics, thermodynamics) can be applied. The big mass has a velocity of $n$ times that of the small one (upon hitting the water surface). $M=nm$, where $M$ is the value of the big mass and m that of the second mass.
We make the small mass fall into the water, record the whole process, and replay the record in slow-motion (suitably adjusted to the situation; let's say that we replay the record n times as slow, but maybe other paces are better). Is one able to see the difference with the big mass falling into the water in real-time?
Is the process of a mass that falls in water scale-invariant wrt the slow-motion version of a smaller mass falling into the water?
In the definition of scale-invariance, the variation of time isn't spoken about, though (see here):
In physics, scale-invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
Time can't be varied in reality (it's a parameter). So strictly speaking, we can't talk about scale-invariance. I think it's clear what I mean though.
I make this edit because something has become clear to me. Of course, it's the splash that's problematic. In a slow-motion version of a small stone hitting the water, the water will have less time to evolve. And of course, there is less water to evolve. This means that one can tell if the splash is caused by a small mass or a big mass (as one can easily tell that a slow-motion trick is used in old disaster movies). And even though a slow-motion replay of a recorded droplet falling in water looks globally the same (initially; later on a high central uprise of water is to be seen and I'm not sure if water is uplifted in the same way for a big rock), one can tell the size is small. I don't think that changing the speed of impact changes much. Upon replaying the slowed-down speed must resemble the speed of the big mass when it hits the water (which can be done by placing (spacially) scaled down familiar objects around the water.
When one tries to model ocean waves (like in MASK) or in this real model of a rogue wave) the overall result looks good but nevertheless one can see immediately that the waves are scaled-down (like in the small droplet case).
So maybe we can use a liquid that's visibly the same as water but with different properties. The surface tension of water plays a role of significance in the development of the splashing water. Maybe using a liquid with a small surface tension renders a more faithful image of real water when the record is played in slow motion.
For the rock, the water has more time to evolve than the water (transparent liquid) for the droplet. This clearly affects the water (liquid) development but I can't visualize exactly how.
So maybe it's best to just go to a lake and let a rock fall into it while not forgetting to record it. Can this falling rock record on its turn, when replayed in slo-mo be used to mimic the impact of a 100 meter asteröid...?