Re: Force required for a vertical jump
I think these responses illustrate the point that that simple questions usually have complex answers.
All of these points are good ones, and I highly recommend doing the math to estimate the answer. However all of these answers are still estimates because the only way to really know the answer is to measure it. You might think that a relatively simple setup with a force measuring device would give you an answer, but there are still many external factors to account for - many of these factors are "human" factors ... a subject who is confident will probably jump differently to someone who is not sure that they can do it, someone who has practiced will probably be very efficient and use less energy than someone trying it for the first time.
A very confident and agile subject might use a lot of force, jump and clear the box by half a metre before landing, while someone who is worried that the box might fail when they land on it, or that they might fall off the box as they land, will jump very cautiously with less force. If the box is quite low, a few centimeters, then the force required will be low - and we would expect that more force will be needed as the box height increases, but if subject is worried that the box height is close to the limit of their ability, they may drastically increase the force to make sure that they clear the top and land without tripping.
So even the simple act of measuring has complications.