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To travel a certain distance s, is it more energy efficient to use one massive jump, or several small jumps?

I tried to solve this problem I posed myself by assuming energy spent on a jump was proportional to the force used during that jump, and that a jump was simply defined as a constant force for 0.1 seconds on the ground.

$$ Energy \; used \;\alpha \;Force \;used $$

From here, the force used in a jump would be proportional to the acceleration, and therefore proportional to the initial speed of the jump.

$$ Force \; used \; \alpha \;Initial \; speed $$

From here, I proved that to travel the furthest, the best angle would be that of 45 degrees(pretty simple proof), so there really is only one angle jump to use to be energy efficient, no matter if you are using small jumps or large jumps. Cool. After some more slogging of equations, I find that the distance traveled (ignoring air resistance) is proportional to the initial speed squared.

$$ Distance \; traveled \; \alpha \;Initial \; speed^2 $$

Therefore, going back up through the equations, we can see that

$$ Energy \; used ^2 \; \alpha \;Distance \; traveled $$

From this, I can conclude that for a given distance, one big jump is more energy efficient than several small jumps. Lets take an example. Lets assume that I have a energy unit Joshe, which means $$ (Energy \;used)^2\;joshes = (Distance\;traveled)metres $$

And the distance to travel is 100 metres. If I do it in one big jump, then I only use 10 joshes of energy. If I do it in 10 small jumps, then I have to use $10\sqrt{10}$ joshes of energy, which is much larger than 10 joshes.

My question is, firstly did I get my calculations correct, and secondly if there are any real life calculations on this topic, because I know I ignored a lot of factors in my calculations. Like a lot. For example, for a human at least, taking several small jumps (at least in my opinion when I do some testing), seems to take less energy than one massive jump. And of course, we do have limits on how far we can jump, and all that. And is energy used even proportional to force exerted? It seems logical but I wouldn't really know.

EDIT: Major "What did I just do" moment. This approach is probably a lot better. Firstly, the energy consumed by our body is proportional to the kinetic energy we gain, and therefore proportional to our initial speed squared. $$ Energy\;used\;\alpha\;Initial\;speed^2 $$ And as we proved above, initial speed squared is proportional to distance $$ Initial\;speed^2\;\alpha\;Distance\;Traveled $$ Therefore: (drumroll) $$ Energy\;Used\;\alpha\;Distance\;Traveled $$ Hang on... Doesn't this mean, IT DOESN'T MATTER IF I TAKE SMALL JUMPS OR ONE LARGE ONE??? To travel 100 metres, with one large jump, it costs 100 joshes of energy, with 10 small jumps it costs 10*10 joshes of energy, or 100 joshes of energy.

To travel a certain distance s, is it more energy efficient to use one massive jump, or several small jumps?

(First approach deleted, didn't make sense) This approach is probably a lot better. Firstly, the energy consumed by our body is proportional to the kinetic energy we gain, and therefore proportional to our initial speed squared. $$ Energy\;used\;\alpha\;Initial\;speed^2 $$ Then, I managed to prove that the angle at which the largest distance is covered given a initial speed is 45 degrees, meaning that all jumps, regardless if you are going for one big one or several small ones, are all at 45 degrees. Then by mashing through equations I found initial speed squared is proportional to distance.(checked on wiki: http://en.wikipedia.org/wiki/Range_of_a_projectile) $$ Initial\;speed^2\;\alpha\;Distance\;Traveled $$ Therefore: (drumroll) $$ Energy\;Used\;\alpha\;Distance\;Traveled $$ Hang on... Doesn't this mean, IT DOESN'T MATTER IF I TAKE SMALL JUMPS OR ONE LARGE ONE??? Now here, as an example, lets just introduce this fictional measure for energy, the Joshe. We can say $$ (Energy\;Used)joshes\; = \;(Distance\;Traveled)metres $$

To travel 100 metres, with one large jump, it costs 100 joshes of energy, with 10 small jumps it costs 10*10 joshes of energy, or 100 joshes of energy. Thus I can conclude that in an ideal world, it doesn't matter if you use one big jump, or several small jumps, to cover a distance, they all use the same amount of energy? Or not? There are several things I have ignored, and in practice, it is impossible to just simply jump however far you want. So my question is, is this a good approximation? If not, in real life, is it actually better to take a big jump or several small ones.

To travel a certain distance s, is it more energy efficient to use one massive jump, or several small jumps?

I tried to solve this problem I posed myself by assuming energy spent on a jump was proportional to the force used during that jump, and that a jump was simply defined as a constant force for 0.1 seconds on the ground.

$$ Energy \; used \;\alpha \;Force \;used $$

From here, the force used in a jump would be proportional to the acceleration, and therefore proportional to the initial speed of the jump.

$$ Force \; used \; \alpha \;Initial \; speed $$

From here, I proved that to travel the furthest, the best angle would be that of 45 degrees(pretty simple proof), so there really is only one angle jump to use to be energy efficient, no matter if you are using small jumps or large jumps. Cool. After some more slogging of equations, I find that the distance traveled (ignoring air resistance) is proportional to the initial speed squared.

$$ Distance \; traveled \; \alpha \;Initial \; speed^2 $$

Therefore, going back up through the equations, we can see that

$$ Energy \; used ^2 \; \alpha \;Distance \; traveled $$

From this, I can conclude that for a given distance, one big jump is more energy efficient than several small jumps. Lets take an example. Lets assume that I have a energy unit Joshe, which means $$ (Energy \;used)^2\;joshes = (Distance\;traveled)metres $$

And the distance to travel is 100 metres. If I do it in one big jump, then I only use 10 joshes of energy. If I do it in 10 small jumps, then I have to use $10\sqrt{10}$ joshes of energy, which is much larger than 10 joshes.

My question is, firstly did I get my calculations correct, and secondly if there are any real life calculations on this topic, because I know I ignored a lot of factors in my calculations. Like a lot. For example, for a human at least, taking several small jumps (at least in my opinion when I do some testing), seems to take less energy than one massive jump. And of course, we do have limits on how far we can jump, and all that. And is energy used even proportional to force exerted? It seems logical but I wouldn't really know.

EDIT: Major "What did I just do" moment. This approach is probably a lot better. Firstly, the energy consumed by our body is proportional to the kinetic energy we gain, and therefore proportional to our initial speed squared. $$ Energy\;used\;\alpha\;Initial\;speed^2 $$ And as we proved above, initial speed squared is proportional to distance $$ Initial\;speed^2\;\alpha\;Distance\;Traveled $$ Therefore: (drumroll) $$ Energy\;Used\;\alpha\;Distance\;Traveled $$ Hang on... Doesn't this mean, IT DOESN'T MATTER IF I TAKE SMALL JUMPS OR ONE LARGE ONE???

To travel a certain distance s, is it more energy efficient to use one massive jump, or several small jumps?

I tried to solve this problem I posed myself by assuming energy spent on a jump was proportional to the force used during that jump, and that a jump was simply defined as a constant force for 0.1 seconds on the ground.

$$ Energy \; used \;\alpha \;Force \;used $$

From here, the force used in a jump would be proportional to the acceleration, and therefore proportional to the initial speed of the jump.

$$ Force \; used \; \alpha \;Initial \; speed $$

From here, I proved that to travel the furthest, the best angle would be that of 45 degrees(pretty simple proof), so there really is only one angle jump to use to be energy efficient, no matter if you are using small jumps or large jumps. Cool. After some more slogging of equations, I find that the distance traveled (ignoring air resistance) is proportional to the initial speed squared.

$$ Distance \; traveled \; \alpha \;Initial \; speed^2 $$

Therefore, going back up through the equations, we can see that

$$ Energy \; used ^2 \; \alpha \;Distance \; traveled $$

From this, I can conclude that for a given distance, one big jump is more energy efficient than several small jumps. Lets take an example. Lets assume that I have a energy unit Joshe, which means $$ (Energy \;used)^2\;joshes = (Distance\;traveled)metres $$

And the distance to travel is 100 metres. If I do it in one big jump, then I only use 10 joshes of energy. If I do it in 10 small jumps, then I have to use $10\sqrt{10}$ joshes of energy, which is much larger than 10 joshes.

My question is, firstly did I get my calculations correct, and secondly if there are any real life calculations on this topic, because I know I ignored a lot of factors in my calculations. Like a lot. For example, for a human at least, taking several small jumps (at least in my opinion when I do some testing), seems to take less energy than one massive jump. And of course, we do have limits on how far we can jump, and all that. And is energy used even proportional to force exerted? It seems logical but I wouldn't really know.

EDIT: Major "What did I just do" moment. This approach is probably a lot better. Firstly, the energy consumed by our body is proportional to the kinetic energy we gain, and therefore proportional to our initial speed squared. $$ Energy\;used\;\alpha\;Initial\;speed^2 $$ And as we proved above, initial speed squared is proportional to distance $$ Initial\;speed^2\;\alpha\;Distance\;Traveled $$ Therefore: (drumroll) $$ Energy\;Used\;\alpha\;Distance\;Traveled $$ Hang on... Doesn't this mean, IT DOESN'T MATTER IF I TAKE SMALL JUMPS OR ONE LARGE ONE??? To travel 100 metres, with one large jump, it costs 100 joshes of energy, with 10 small jumps it costs 10*10 joshes of energy, or 100 joshes of energy.