How fast can water from a lake be frozen? Warning: this question contains spoilers for the first episode of 2018 TV series Lost in Space by Netflix.
On the series, the Robinsons spaceship, Jupiter, falls on a planet, and it’s submerged in the water of a glacier that melted. Judy enters the water and swim to the spaceship to get a battery. A minute later, she can’t get out because the water freezes too quickly, holding her on the ice. 
Accordingly to what was said on the episode, we know:


*

*The planet's environment: "earth-like atmosphere, air-pressure and gravity"

*The lake's depth: "once the water freezes, the Jupiter's gonna be locked in 50 feet of solid ice."

*The temperature: "in six hours, the sun's gonna go down, and indications are, it will drop to 60 degrees below zero"


So, given this information, I would like to know if it’s possible for water to freeze so quickly like that?
Note: I’m no expert on physics, so I appreciate if you could use a not so technical language.
 A: As we don't know anything about the intial temparature of the water, another fun but unlikely explanation would be to consider what we call supercooled water.
Basically, at atmospheric pressure, water below 0°C doesn't have to be solid. Sure enough, it would be its most natural state. But actually if you cool down some very pure, very steady water, you can pretty easily get... a very cold liquid below 0°C. 
This is all because in order for the water molecules to re-arrange into Crystal of ice, they need some support to rely on, a little seed to grow around (we call that a nucleus). So if there is no impurity in that water, then you can cool it down without freezing it ! You have to be careful to keep it steady too, because the energy of a shock would give the kick it needs to start the crystallization.
You can find plenty of YouTube videos demonstrating this phenomenon.
So if we assume that your water is already below zero, and in a supercooled state, then the swimmer in it would kickstart the crystallization, which would happen very quickly. For all I know, it is plausible that she'd then get trapped Under ice.
Your stroy makes me think of that because there is a horrible but pretty cool legend about a lake in Russia called the Ladoga lake. Alledgedly, during the Siege of Leningrad in 1942, in a cold winter's night, a fire started in the forest, scaring off a thousand of horses. Those who could run away from the flames plunged into the cold waters of the lake. And suddenly, with a sound of broken glass, the lake froze as the horses were jumping in it, trapping hundreds of them them in ice.
I think it's Hubert Reeves who first suggested that the lake might have been in supercooled state, which could explain this sudden freezing, with the carateristic sound of shattered glass.
However most people suggest it's just bullshit, as the water would have to be insanely pure and steady (which is not very natural).
Furthermore, I have never seen an experiment about supercooled wtaer on such big scales. I don't know if this would hold.
So in short, supercooled water might be an (unlikely) way to justify it. But not so bad regarding series standards :)
A: I just finished watching the pilot episode and the way they treated the physics of freezing water alone is enough to make me not want to watch the second episode.
First I will say that I am not a physicist by any means, although I have been interested in physics from a young age.  My expertise stems from the fact that I have lived most of my 42 years in Minnesota, and I can say that I have learned a thing or two about how water behaves in sub-zero temperatures.
Others have pointed out that the water could have been supercooled and only needed a nucleation point to freeze.  I hold that Judy wasn't the first agitator of the supercooled water.  Have we all forgotten that a ship just crashed into the glacial lake? That should have been enough of a nucleation point to trigger the crystallization.
Unless there is a source of cold coming from below the surface of the planet (which there doesn't seem to be because several miles away from the arctic landscape is a temperate coniferous forest) then the water would, as others have pointed out, freeze from the top.  If it was cold enough outside to freeze the water to a depth of even a few inches in that short amount of time, the people on the surface would have frozen much faster, and not been able to do anything with exposed skin.  Furthermore how could it possibly rain when the ambient temperature is -60 degrees.  I don't think they say whether it's Fahrenheit or Celsius, but I would argue that it doesn't matter.  -60F and -60C are both extremely cold temperatures, and enough to freeze exposed skin in a matter of minutes. While the rain continues to pour down as a liquid, it freezes when it hits the lake.  Why doesn't the rain freeze on the peoples' faces or on the screen of the (waterproof?) laptop?
It is clear that the writers have not spent any amount of time in cold places and use water and ice as convenient plot devices without any regard to physics.
A: More information would be needed in order to do what you want. To calculate the time it would take for the lake to freeze, we would first of all need to find the energy that must be lost by the lake in order to freeze it completely, which we can't do unless we know the superficial area of the lake. (We must know how much water we're freezing). We also don't know what the initial temperature of the lake is! Therefore, we can't even  know how much energy should be spent  to freeze even one unit  of volume.  If you manage to estimate those missing quantities, then you must face the challenging task of finding the rate at which heat is lost by the surface. For this last part you should search the internet, since the rate at which energy is lost on the surface of a lake has probably been modeled by someone, given that there are practical applications to this problem. But then again, this won't solve your problem without the missing information.
Edit: Since the situation only took one minute, you could maybe assume that she entered the water when it was already very close to the melting temperature, unless there is a reason for the temperature in that planet to change very much in a matter of seconds. This way you could eliminate the (otherwise impossible to obtain) initial temperature and assume the total energy is that needed to freeze water at its freezing temperature. It also makes whoever attempted it very incompetent for not noticing this, but I digress.
