Rain, hail, and speed

The physics of falling things

(from Night Cafe — prompted by the Author)

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This time Chloe’s meeting was called under a balcony, sheltered from the rain (for those who don’t know, Chloe is the kitten who allows S. and me to share her spaces and passion for Physics). I did not have indications of any specific topics; one thing: a couple of days ago, I saw her sleeping on a page of a statistics book, and we know that cats, when they are sleeping… are probably doing something different.

The attendants are arriving with the usual feline nonchalance: I recognize some of them. While they wait, they are doing usual things (licking their fur, getting their nails done, or sleeping), until a gray cat makes a big stretch and says:

“Hi friends, given this winter, I asked Chloe to tell us a little about the rain, but let it go beyond the usual things.”

“Good idea! Can I start? — says a red-striped kitten — Is it true that in the high mountains the raindrops hurt less because the clouds are lower and the drops travel less space, so they have less speed?”

“Who the hell told you this?” says a brindle cat, stopping licking his paw for a moment. I look at Chloe: she’s rolling on her back, and I suspect she’s laughing…

“You could just explain it, instead of making fun of me,” says the red kitten, “after all the things you told us about kinetic and potential energy… They show me about mountain villages where the clouds are nearer than at the sea.”

Cloe recovers: “I wasn’t kidding you, I was looking for inspiration to answer.” She pauses for a moment, then: “You see, you’re right: the height of a cloud is always measured relative to the sea. So if a cloud is 3000 m above the sea, the same cloud in a place at 2000 m of altitude will be at 1000 m above the ground. Hence the reason for what they told you…”

Red Kitten turns towards Tiger Cat, blowing (it looks more like a raspberry to me, to be fair).

“But… — Chloe continues — Whoever told you this, forgot that rain falls through the air and therefore suffers viscous friction. So, the drop begins to fall, and as it falls, its speed increases, but friction increases too, because it depends on that speed; this continues until the two forces equal each other:

The friction increases with speed (and time)

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At that point, the drop, no longer subjected to a net force, moves at a constant speed!”

“And when will that happen? If it takes that long, then I was right. Just to see if it’s worth going to the mountains,” asks Red Kitten stubbornly.

The linear friction

“To answer this question, we need to build a mathematical model. Let’s start by assuming that:

• The drop can be considered almost spherical.
• Its speed is low.

Then the friction can be evaluated by Stokes’ law:

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where 𝜂 = 1,8 ∙ 10⁻⁵ Pa·s, air viscosity, r is the drop radius. The gravitational acceleration g can be assumed constant over a difference in altitude of just 3000 m, so the weight is also constant, and the equality of the two forces leads to

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Assuming the drop radius of 1 mm, its mass is obtained from the density 𝜌 of the water and its volume:

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Doing the math, the speed limit is around 120 m/s… Nobody notices anything?” “Of course — a three-colored kitten says — you just assumed a small speed…” “Very good!” Chloe wags her tail. “The speed found is too high, not consistent with the approximation made!”

The quadratic friction

“So — goes Big Grey — how do you get out of it? Should we use quadratic?” “Exactly — replies Chloe — we must assume that friction is given by

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A = 𝜋r² is the surface shown to the air, 𝜌ₐ= 1,2 kg/m³ is the air density, and Cₐ = 0,5 for a sphere. Now the limit speed becomes

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Of course, wind and air currents can change this average value.”

“But to answer the question, we still lack something: when is this speed reached?” points out the three-colored kitten.

“You’re right. We need to check whether the space is sufficient to reach this speed, which is not a simple thing, because the acceleration is not constant: it starts from g and reaches zero; we need to find how the speed varies as a function of time, v(t); for those who want to see what the mathematical model is, we can apply F = ma:

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To simplify this differential equation, it is advisable to insert the limit speed formula:

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I skip the usual math steps and get the speed as a function of time:

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The constant 𝝉 shows how quickly the speed approaches the limit; for our data, 𝝉 ~ 0,7 s. Within 2 to 4 seconds, the limit is reached. So the speed at land level is the same both at sea and in the mountains!”

v(t): speed as a function of time

“When I see that guy, I give him something that he’ll remember forever…” says Red Kitten.

“What happens with hail instead?” asks a white cat with a lower ear, so far silent. “That really hurts. I caught one of them while I was in a meadow…” “It doesn’t change much: compared to a drop, the coefficient Cₐ is a little lower, so the limit speed increases, but not so much: if you do the math, you’ll find about 8 m/s”. “I heard humans worried about hail falling on one of their tin boxes they call ‘cars’…” says the Red Kitten.

“Meeeooow! But that’s another story, it doesn’t depend on the speed difference!” Chloe meows. “Of course — says Red — it’s because hail weighs more and therefore can do more damage.” “How silly you are — Grey adds — everyone knows that ice is lighter than water.” “Then I’ll throw you a ball of ice and see if you like it,” Rosso says in a flash. “Calm down, please — Chloe shouts — It’s true, ice is lighter than water, but this isn’t just about mass.”

The Momentum

“Meeeow, momentum?” says Three-Colored. “But she just told you it doesn’t depend on the speed differences,” says Grey. “True! It depends on both mass and velocity, that is, momentum, as she said!”. Three-Colored yawns, pointing out how obvious it was. “Do you remember that a kitten going fast does the same damage as a big cat that goes slowly?” “Get me going, and I’ll fix you all, meow!” warbles Red.

Chloe moves to an area with wet earth and begins drawing on it: the others approach, as if there were food somewhere.

“Do you remember that the law of dynamics can also be written as

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That is, the instant change in momentum gives the external force. In our case, it’s the horizontal plane that exerts this force to stop the drop: this is the same force made by the drop on the plane. Suppose that the drop and the ice grain have the same mass, which is quite true

When the droplet, position 1, falls onto the horizontal surface, it usually changes shape, perhaps splitting into a thousand droplets: in general, the water remains on the surface, position 2; the impact of the droplet is completely inelastic. If m is the mass of water and vg its velocity before reaching the ground, the force exerted by the droplet on the plane is

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Now let’s consider the hail grain; it is somewhat elastic: when it hits the surface, it doesn’t break down, but bounces back, position 4:

We don’t know the speed after bouncing, we can only say that it is less than the previous falling speed; some energy is lost in the impact, maybe in the noise it makes. Let’s use our experience: when hail falls, how much does a hailstone bounce?” “About two legs,” replies Grey. “I think it’s at least three legs,” says Red. “Okay, translated, the bouncing could be about 5 cm; then, from the conservation of mechanical energy, neglecting air friction…” “But we didn’t neglect it before; we got the limiting speed. Can we consider the energy as conserved?” Three-Colors asks. “Not in the impact, but in what happens immediately afterward, yes: here the speed is low and the distance very short. Therefore, doing the math, kinetic energy at the bottom and potential energy at the top (h = 5 cm), conservation tells me that the bouncing speed is

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The force exerted by the hail grain is

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“And how do we find 𝛥t?” asks Three-Colors. “We only know that it is very small, as in all impacts, but we can think that it is the same for both the drop and the grain. So let’s make the ratio between the two forces: it does not depend on the time interval:

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Here’s why the grain hurts more than the drop: the force made on the plane is 50% higher for the grain!”

The attenders remain silent; Chloe adds: “Hence the feeling of pain when a grain hits you. Also note that the formula doesn’t contain the mass: the ratio depends only on the velocities.” “The ratio. — says Three-Colored — But the forces do depend on the mass.” “Sure, a 10-grams grain is very bad for you, but always 50% more than a 10-grams drop.” “So, — Grey says — we confirm that it’s best not to wander around under the hail…”

“M(eee)ooooww… I’d like to know if it’s good to run when it rains…” says Red. “This is fun, but it’s also a little long. Let’s talk about it next time. We have discussed enough for today. If you want, you can stay safe here, but I’m going: I can smell that my human is preparing chicken, and if I were cautious, I can take a piece of it away… and I’ll also get a little exercise, which I really feel the need for… Meeow!”