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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...?

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    $\begingroup$ Are you only interested in the fluid dynamics aspects after impact? You will notice the difference in slow motion if you record the free fall ("moon effect"). If you consider aerodynamic drag as well, absolute mass scale comes into play (smaller feathers ten times the mass of bigger feathers behave different than small "rocks" ten times the mass of big "rocks"). Also, will the falling objects have surface structure that give hints to their composition/density? Is there something in the movie that indicates length scale? Will the image crop be the same for both videos? $\endgroup$
    – oliver
    Mar 20, 2021 at 12:39
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    $\begingroup$ I think if you want the splash to look the same in both cases, you need to use a lower viscosity fluid for the smaller version. If you use water, it will look like a big ball falling into syrup. $\endgroup$
    – Ben51
    Mar 20, 2021 at 14:40
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    $\begingroup$ I am wondering why you didn't Post charts/graphs/pictures showing what you meant? $\endgroup$
    – Robbie Goodwin
    Mar 21, 2021 at 0:01
  • $\begingroup$ @RobbieGoodwin I will do so tomorrow. Right now my bed (and my two girls) are waiting for me. :-) $\endgroup$
    – Deschele Schilder
    Mar 21, 2021 at 0:17
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    $\begingroup$ I know avalanches are modeled by building a miniature of the mountain in question, piling up little plastic balls on the slopes, surrounding it with water, and setting it going. That sounds mildly related $\endgroup$
    – Arthur
    Mar 22, 2021 at 18:31

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In a crude way, the answer is yes, and this effect was commonly exploited in old school sci-fi films by shooting scaled-down models and playing back the action in slo-mo.

This general effect is also exploited in the world of physics and engineering, where accurate scaling factors are used to model the behavior of full-scale systems by using models in which scale lengths, speeds, accelerations, forces and whatnot have all been scaled together through the use of nondimensional similitude parameters which will properly proportion all the pertinent effects. There are dozens of these, each with its own special area of application, typically named after the researcher who first described their use.

Common examples include the Reynolds number, Mach number, Nusselt number, Froude number and my favorite one: the Number number, which tells you how many nondimensional scaling parameters are needed to solve a particular problem. This is an engineering joke, you may laugh now.

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    $\begingroup$ What actually prompted my question was the use of some liquid (with a higher viscosity than water) in the modeling of huge water waves (super tsunamis). I wondered if one could simply use normal water, video record the wave, and replay the record in slow motion. What also prompted it was the use of a huge Titanic model in the eponymous movie. The movement of the water was, later on, computer-generated. Real models in water disaster movies always seem so unreal (but very funny though!). $\endgroup$
    – Deschele Schilder
    Mar 20, 2021 at 18:29
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    $\begingroup$ "typically named after the researcher who first described" —not quite: see Stigler's law of eponymy. $\endgroup$
    – Ruslan
    Mar 20, 2021 at 19:46
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    $\begingroup$ @DescheleSchilder The problem is that different numbers scale differently. That is okay if you want to model a phenomenon associated with only one of them. For aerodynamic drag of a car, you pick a scaling with the same Reynolds-number, for the drag of a boat the Froude-number, for shock waves you need Mach and so on. But to accurately model a chaotic situation involving all of them, your only choices are life-sized models or detailed computer situation. That's why old disaster movies look so unreal, as neither was an option, so they just picked a scaling that looked spectacular enough. $\endgroup$
    – mlk
    Mar 21, 2021 at 13:31
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    $\begingroup$ @nielsnielsen Which one? The professional basketball player, the MD who specializes in physical rehabilitation, or the one recorded on the Michigan Offender Tracking Information System? All found on the first page of Google. $\endgroup$
    – nick012000
    Mar 22, 2021 at 4:58
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    $\begingroup$ Regarding the filming technique: The movie Downsizing has a clever example of the inverse of this effect. This movie, in which humans are shrunk to Barbie-scale, has a scene on a boat. They obviously shot the scene on a real boat but in post-production, re-engineered the water to make it look bulky and viscous, so that it behaved as water would around a real 50cm model boat. $\endgroup$
    – Oscar Bravo
    Mar 22, 2021 at 6:59
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We can apply Dimensional analysis to answer this question. The obvious relevant constants are

  • density $\rho \propto \frac{M}{L^3}$
  • free fall acceleration $g \propto \frac{L}{T^2}$
  • viscosity $\eta \propto \frac{M}{LT}$

With these two, we can simultaneously scale size $L$, mass $M \sim L^3$ and time $T \sim L^2$ while keeping density and acceleration constant. All physical processes depending only on them will occur in exactly the same way.

However, there are some subtle physical processes in your system which involve more constants:

  • surface tension $\sigma \propto \frac{M}{T^2}$
  • speed of sound $v_S \propto \frac{L}{T}$

With them included, we can no longer scale the system while preserving the constants.

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    $\begingroup$ Yes, surface tension is the give-away, I would say. A scaled-up, slowed-down version would look too smooth to be real. $\endgroup$
    – TonyK
    Mar 21, 2021 at 13:18
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    $\begingroup$ @TonyK - That's often seen in old films about pirates and naval battles, where water surface gives away that the ships filmed are just small scale models. $\endgroup$
    – Pere
    Mar 21, 2021 at 20:24
  • $\begingroup$ Surface tension is almost certainly going to have a bigger effect than viscosity, actually. $\endgroup$
    – Buzz ♦
    Mar 22, 2021 at 1:44
  • $\begingroup$ Is it the surface tension of fuel that accounts for the innaccurate look of firey explosions in Gerry Anderson's Supermarionation productions? I believe that he's responsible for several generations of children growing up thinking that explosions look wildly different from the way they do in reality, because the grew up with his miniaturised slow-mo explosions :D $\endgroup$
    – Brondahl
    Mar 22, 2021 at 7:40
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No

The water droplets have a certain size no matter the size of your stone, so if you record a small stone the water droplets will look larger relative to the stone, than if you recorded a big one.

Also, the amount of droplets will be different, due to the energy differences as explained above.

I still think it could be worth a try, even though it doesn't scale perfectly it might be good enough.

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The energy of the impact is proportional to the mass (for the same velocity). And $M \propto R^3$. But the area that receives the impact is proportional to $R^2$.

So, I suppose that the splashing effects increases with $R$, (energy by area).

One possibility is to increase the speed of the falling small sphere until get the same ratio. And adjust the speed of the film to the slow motion required for a realistic free falling velocity.

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No. The reason is the well-known Square–cube law. In fact, you have to deal with a linear-square-cube law: You'll have quantities that scale linearly (the 10 cm model will move with the apparent speed of the 1 km asteroid through your video), you'll have some that scale with the square (inner cohesion of bodies, repulsive forces in collisions), and you'll have quantities like mass and hence kinetic energy that change with the cube of the scale.

For example the mass and hence the kinetic energy at a given speed of a model at a scale of 1/10 will be 1/1000. Say you want to film a collision with a speed of 10 m/s of two moons measuring 1000 km in diameter by colliding two marble balls. The moons at that speed will deform and potentially fuse together; the marble balls may perform an elastic collision and bounce off of each other. Their inner cohesion is large compared to the masses and energies.

You may be able to increase the kinetic energy of the balls — until it is large enough to overcome their inner cohesion — by letting them collide at higher speeds, but you'll see things you don't see with the slower moon collisions. The reason is that the inner cohesion of the marble balls is significant; celestial bodies, by contrast, have only negligible inner cohesion: They are essentially plastically deformable matter held together by gravity. Their collision will be non-elastic, like putty. A high-speed collision of marble balls, by contrast, will resemble an explosion.

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