Big jump or small jumps?

Question

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.

Answer 1

Looking at real life jumpers, Kangaroos, the key element seems to be the efficiency with which the jump energy can be recovered upon landing. This is usually done in all animals, including Humans, by storing energy in the connective tendons.

Answer 2

Your purely mechanical considerations are correct.

When considering animals jumping and taking into account the <100% efficiency of the muscles converting chemical energy to the kinetic one, realize that:

• Making lots of very tiny jumps is definitely very inefficient, as the animal burns a lot of energy for just keeping the position, basal metabolism etc.
• Making the largest jump one can do possibly leads to a muscle damage which is also very energy inefficient if the energy needed to regenerate is included.
• It seems reasonable to assume that the dependence of the efficiency on the jump length is a continuous function. Hence, this function probably looks like a kind of bump and, thus, there is an optimal length of a single jump but its value definitely depends on the animal species considered.