# Muscle mass required for organisms to jump

If an organism weighing roughly 20kg existed (same weight as a dog), similar also to the structural composition a dog. How much muscle mass would be required within those legs for it to jump 20m?

Dogs can jump up to 1.8m, their rear legs being around 10cm in diameter. Assuming the legs are half muscle, half bone the cross sectional muscle will be around 5cm. If the theoretical organisms muscles were around 20cm in diameter and rear legs were about 80% of their mass, how far could they jump?

Dogs use the method of direct muscle contraction as their method of locomotion, this is not effective for vertical jumps. Grasshoppers on the other hand employ a type of catapulting by the use of specialized cuticles in their rear knee joints. Through using this method they are able to gain vertical momentum with much more efficiency and therefore jump much higher. If the theoretical organism were to employ this type of locomotion instead while retaining the same amount of muscle mass how far could it jump?

• Are you trying to jump long or tall. There's a fascinating argument I came across recently that jump height is almost indepentent of creature sizes. If you look at everything from a flea to a dog to a human (a range of 10^8 in mass), they all jump to within 1 order of magnitude (20cm to 2m). If you're looking for a 20m high jump, the answer may be that it simply cannot be done, or if it can be done it will require a complete redesign of the body structure to do this task. – Cort Ammon Mar 14 '18 at 19:34
• That is an interesting argument. The jump does not need to be completely vertical, it could even be only 5-10m. Originally I looked a plant hoppers but after some research I found it would be impossible to have them on a 20kg scale due to the cube law, the larger they become the less strength their muscles have in comparison to their weight. Ideally the organism would be about the size of a dog, but at this size it would be impossible for them to jump to such a height through the use of a dogs direct locomotion, that's where the grasshoppers specialized legs come in. – Lutro Mar 14 '18 at 19:43
• Yes, the square-cube law is the bane of many a question on this site. :) Welcome! – a CVn Mar 14 '18 at 19:55
• I want to upvote this question, but I've reached my voting limit for today :\ I'll be back in a couple hours. – Renan Mar 14 '18 at 21:26

This topic really boggles my mind. It is one of the few placed where the square-cube law doesn't rear it's ugly head!

The first thing to note is that we are only looking at animals that can jump. Nobody cares that the millipede can't jump at all. Elephants can't jump either, which is due to the way their body is structured to deal with the square-cube law. We're looking at animals that do jump, so we assume that their body has evolved to optimize that sort of motion to at least some degree.

The most obvious limitation is power. The maximum power of a muscle is proportional to its cross-sectional area. If you don't have the power to emit energy quickly, you can't use it to jump. In practice, however, this isn't a limit. Creatures that can jump are typically structured in a way that creates enough power. For humans and dogs, that's accomplished by having enough of a cross-sectional area in the legs to do the job. For the click-beetle, it's a bit more difficult. Their muscles aren't big enough to provide this sort of power. To work around this, the click beetle actually bends its shell using its muscles, then releases all of that energy at once. This gives the beetle more time to put energy into the system from its muscles, circumventing the power limits. So as it turns out, power is not the primary limiting factor.

Another factor that shows up is drag. Animals like click beetles and fleas actually get limited by drag, so they jump to lower heights than the larger animals do. However, you are interested in a dog sized animal, so this wont matter.

The dominating factor is actually energy. To jump, you need to accelerate upwards, and that takes work. The more work you can do, the more upward velocity you can have. It turns out that the energy you can exert scales proportionally with muscle mass. A muscle can only output so much energy per unit of muscle-protein, limited by the chemistry of how muscles contract. Once a muscle is fully contracted, you can't contract it again until it gets to relax and elongate, and by that point the jump is already over.

So we have a curious result: jumping energy scales proportional to mass, and the amount of energy taken to jump to any given height scales proportional to mass (potential energy = mgh). Jumping height is not proportional to mass at all! This highly unintuitive result can be seen in the table below, which is copied from a text book from which this entire argument is paraphrased (this, itself was found on a previous StackExchange answer)

• Flea - $5\cdot10^{-4} g$, 20cm jump
• Click Beetle - $4\cdot10^{-2} g$, 30cm jump
• Locust - $3\cdot10^0 g$, 59cm
• Human - $7\cdot10^4 g$, 60cm

Despite the fact that those masses span a range of $10^8$, jump height stays within 1 order of magnitude!

Dogs are no exception. The current record jump is 2-3m, depending on what kind of jump we are talking about. They too fit in that 1 order of magnitude, or perhaps a little more.

So what have we learned? For a vast variety of body plans, jumps remain in the 20cm-3m range due to energy limits. A 20m jump is quite far outside of this range. It's going to be hard to achieve.

Making their body more leg will help. 80% leg as you suggest may give you more altitude, but it's not going to be a slam dunk (literally). Think about how you jump. You crouch down, right? And when you try to jump high, you use your arms on the way up so that you can use elastic forces to help you jump higher. A surprisingly large amount of your body is actually helping with the jump. It will indeed be more efficient to have 80% leg muscle, given that those muscles are in a perfectly ideal place to do the job, but I don't think it's going to give you a factor of 7-8 increase in jump. Too much of our muscle is already involved in the jump.

A simple solution like the click beetle carapace or the grasshopper cuticles won't help either. The issue is the amount of energy that can be expended with one contraction of your muscles. The carapace and cuticles only help you deal with the power issue associated with low cross-sectional areas.

The solution you need is a ratchet. You need something akin to the click beetle carapace or grasshopper cuticle, but one which permits multiple contractions of the muscle to build up energy. You would contract and relax the muscle perhaps 10 times to slowly build up energy, and then have a trigger to release it all at once.

Your question is timely. I recently just asked a question on biology whether such a structure ever occurs in biology. I am not aware of any animal with such a ratchet. We'll see if it gets an answer. But it may be the thing you need to hit 20m.

• This is not my area of expertise, however, through the example of a biological ratchet would the energy be stored as rotational force? If so, do you think an elastic organ would act as a storage medium? Or would it bare more similarities to the carapace of a click beetle? – Lutro Mar 14 '18 at 22:35
• I wouldn't put a ratchet as being unrealistic. Alien, for sure, but there already exists an insect with toothed gears: The Issus Coleoptratus – Andon Mar 14 '18 at 23:12
• @Lutro The specific feature I have not seen is the storage of energy from multiple contractions of a muscle. There are definitely examples of storing the energy, but I don't know of any which can bend a carapace partially, lock it in place with a paul, relax the muscle and elongate it, bend the carapace further, lock it in place again, etc. That would be the required trick to avoid this energy limit. – Cort Ammon Mar 15 '18 at 0:13
• @Andon It's a tricky thing. From an engineering perspective, it doesn't seem all that unrealistic at all. But from a biology perspective, we just don't see it. It's a curiosity to me. There's something about that process of storing energy from multiple muscle contractions that just doesn't seem to happen. I've got the open question on Biology.SE to go see if there's a case I didn't think of. So far, no hits, but we'll see. – Cort Ammon Mar 15 '18 at 0:15
• @Andon Of interest to me is that those gears in I. coleoptratus you mention never rotate 360 degrees. They only rotate part of a circle, and they rotate via a contraction of the muscles. I. coleoptratus can't engage the mechanism to rotate 45 degrees, stop, reset its muscles, and try again. It just synchronizes its legs with the gears. – Cort Ammon Mar 15 '18 at 0:16