Is it scientifically possible for the hydra to regenerate her lost heads?
Yes, however, many things have sound science, but the practice of it is usually what holds it back, which the other 2 answers will explain.
How long would it take to regenerate one head if it was as big as in the picture above?
This would depend heavily on the last question. Sorry if this answer seems to be dodging the question, as this answer is heavily dependent on energy
Considering the time and energy required to grow a head, could they really grow several heads in a row?
Imagine the human body, it can repair minor injuries, but this requires nutrition from food, and takes quite some time. So, now the hydra needs 2 things, mass and time/energy.
Scaling this up to an entire head of beast of that size, they would at least need mass equal to their lost heads. Note the at least, it is not enough to have 500 kg of protein in your belly to regenerate the head, as the head contains noticeable other things, such as bones, teeth, the brain etc.
So, the Hydra would need to be extremely bloated/fat, containing enough nutrition within it to regenerate it's heads. However, this is great, for your question, as that means that there is an actual scientific end to their head regeneration. If they run out of nutrition, they can not regenerate anymore.
Alright then, let us assume the Hydra is the product of billions of years of evolution, or genetic engineering, or something to that effect. Because of that, the hydra simply needs to provide calories to be able to move the nutrition from her belly to her head and not worry too much about minor things like biology.
Well, you know some of the simple physics equations, Force is Mass * Acceleration, Work is Force * Distance, Power is Work / Time. With those, you can find that depending on how much calories the hydra focuses into the head regeneration, the time taken to regenerate also varies.
Some quick and not too specifically accurate high school level science:
- Suppose that the head weighs 100 kg. (It should really be much heavier, but this is simpler)
- Suppose that the distance from the belly to the head is 5 m. (Do note that this example hydra is much smaller than the one you want, but you should be able to scale it up)
- To regrow the head in 1 s, the amount of force is 100 kg * 5 m/s-2 * 2 = 1000 N. The 2 is to stop the mass at where the head is.
- The work done is 1000 N * 5 m = 5000 J
- The energy required is 5000 J / 1s = 5000 W.
So, the amount of work needed is not too much, only about 1000 calories, but the power output is quite sizable.
However, in the end, the power a creature can generate is largely dependent on it's food, so the limiting thing for your hydra would be how fat you want it to be. Either that, or give it super metabolism and have it eat things with some of it's heads while fighting to fuel it's regeneration