I have often wondered why high jumpers can jump higher if you run. The way I see it is that you only build up horizontal speed, and since you're running on a plane, I cannot see how this speed can be used to increase the vertical speed.
I cannot say that I have any definite proof of this claim, but there must be a reason why ALL high jumpers run instead of just jumping from standing still.
After giving it some thought, and with the help from comments, I figured that running gives you the speed needed to not having to convert any vertical speed to horizontal speed. In high jumping, you need some horizontal speed to get over the pole.
However, the world record for standing high jump is 1.90m, while the record for regular high jump is 2.45. I have a hard time believing that you can gain 55cm purely from this. And especially since standing high jumpers are allowed to perform the jump with both feet. Regular high jumpers are only allowed to have one foot touching the ground during take off.
Best Answer
Having done a fair share of high jumping I would like to share my view on this.
High jumping differs from many other jumping sports, e.g. ski jumping, by the fact that the jump is ballistic instead of elastic. (Not entirely ballistic, but much more so than many other sports.)
The kinetic energy is not entirely conserved in the jump, but some of it is redirected into vertical direction by setting the leg in front of the center of mass. The best high jumpers are the ones who can have the highest velocity, and still not loose to much of the energy when converting the horisontal momentum to a vertical momentum. (Some also seem to have a very strong elastic jump, and don't use as much speed. Typically called power jumpers and not relevant to the discussion.)
In total, how much of an advantage is it to run? World record with approach is $2.45 m$ and the highest I know of without is $1.82 m$.
Edit: For a simplified discussion, I would suggest considering the scissor technique, which does not involve rotating at all, but is without a doubt better than jumping without an approach.
https://www.youtube.com/watch?v=8hz7ZA35U5g Stefan Holm jumping 2.10 m with scissor technique.