Although it's a grim question, I think we can come up with some approximate answers. Firstly, we need to know how easy it is to break bone. For this I had to turn to this research paper by Peter Fratzl on densities and breaking points of biological material. He points out that the angle of impact has an affect, but we're wanting to work out the minimum height required to guarantee bone breakage. In other words we want to take everything at it's least likely extreme. For this, Frantzl measured the impact as needing 9920 J/square meter in order to break a bone.
This tells us how much energy we need for every square meter of bone, so let's work out how much area of bone you've actually got. Unfortunately, nobody seems to have done any good research on this topic so we'll have to make a few estimates. According to Medicinenet.com the average human man has a surface area of 1.9 square meters. We'll use this number rather than the female number, since we're looking for the "most difficult for nature to pull it off" example. We also need to divide this number by two, since all 1.9 square meters don't hit the ground at the same time (you can't hit the water on the front and back of your body simultaneously) so we'll assume you hit the water in a perfect flop, giving you a surface area of 0.95 square meters. This means the water is going to need to apply 9920 x 0.95 Joules of energy to the human body in order to make it break. That's 9424 Joules of energy needed to guarantee a bone break. What height would we need to achieve this kind of impact? Energy done on the body can be calculated as mass x acceleration x distance. So we need the average mass of a human. According to Biomedicalcentral.com the average human has a mass of 62 kg. The average American is 80.7 kg, so let's go with that number. The distance travelled is referring to the distance you travel as you enter the water. This calculation gets extremely complicated because as height above the water changes, so does distance travelled into the water, so I'm going to use another back-of-the-envelope calculation and assume you come to a rest in about 2 meters below the surface. Energy = mass x acceleration x distance travelled. If this is a 9424 Joule change then we need an acceleration of 9424/161.4 (I'm just dividing energy over mass x distance here). This gives us a required deceleration of 58.4 meters per second squared. Acceleration is calculated as (final velocity - initial velocity)/time taken to change. For this calculation, our final velocity is zero (because we've come to a rest) so we need to know how long it takes you to come to a halt once you hit the water. Calculating this is fiendishly difficult because we need to take into account the change in density from air to water, so I'm going to "back of an envelope" it once more and say it takes about 1 second to stop after you've hit the water. If you watch underwater footage of divers this usually seems about right. It also makes the calculation neater because now we can say that acceleration = (0-initial velocity)/1. In other words we can say that the speed you hit the water at is 58.4 m/s. Now here comes the final step. What height do you have to be in order to hit the water at this speed? Well, fortunately for us, all objects accelerate toward the ground at the same rate, so there is a simple formula you can use to calculate final velocity. It's v = sq root (2 x gravitational acceleration x height). We know our final velocity is 58.4, we know gravitational acceleration on Earth is 9.81 meters per second squared which means if we do a simple rearrange we can calculate what height is needed: (velocity squared)/2xgravitational acceleration. In this case that becomes (58.4 squared)/2 x 9.81. This gives an answer of 173.8 meters. So there you have it. If you want to be 99% sure your bones are going to break when you hit the surface of the water, you'd have to jump a distance of at least 173.8 meters. Now, a fall from a much smaller height could still break your bones if you hit the water at a certain angle, but 173.8 is what you'd need to guarantee it. Hope you're happy now Vishal!
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Density is a measure of how much mass you will find in a given volume. Nowadays we define it as the mass divided by the volume but, interestingly, 'twas not always so. In fact, Isaac Newton (the father of modern mechanics) considered density to be the more fundamental property and defined mass as simply the density multiplied by volume. Sam's question is a very good one: at what scale of the Universe do we start to notice density?
In the nucleus of an atom there are three main forces at work. The electromagnetic force causes protons to repel each other, while the strong force causes them to attract, along with neutrons. The weak force causes protons and neutrons to turn into each other and everything is held in a delicate balance. At this scale all three forces play an equally important role and the result is that all nuclei have the same basic layout. All atomic nuclei have a fairly equal spacing of atoms meaning that there is an actual "nuclear density" common to all atoms. The value is around 2.3 x 10^17 kg for every cubic meter of nucleus you have. The space between the nucleus and the electron orbitals also has a pretty consistent density because the distribution of mass is about the same for all atoms. We usually express this in terms of energy per volume rather than mass per volume (because the mass isn't stable) and it has a value of around 1 x 10^-9 Joules per cubic meter. That's quite a lot. Where density starts to vary is above the atomic scale. Atoms are attracted to each other at a great distance but repelled by each other when they get too close. The sizes of these interactions depend on the size of the atom, the shape it has and how well shielded the nucleus is from the outer radius. Different types of atoms will therefore have different ways of squashing against each other. Ultimately this means different types of atoms will pack together differently and density as the real-world property emerges. So density differences are only relevant on the scale of atoms and bigger. Anything smaller and things are nicely balanced. |

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