Can you jump higher if you run? If so, why? (High jumping) 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.
 A: 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. 
A: I don’t profess to know the physics of high jumping, but suggest the following.
It seems the high jumper needs to do two things to clear the bar. 
One is to have horizontal velocity in order to clear the bar and reach the other side. If the high jumper simply stood in front of the bar and jumped up, I don’t see how he/she would get over and to the other side of the bar.
The other, of course, is to gain the necessary height to clear the bar. So the question is, does running give you greater height than the other extreme; say just standing in front of the bar and jumping straight up.  The picture below (which I used for another purpose concerning friction) shows a running with various forces acting upon him and him upon the ground.  The runner exerts a thrust against the ground giving him both a vertical and horizontal component of velocity. It is possible that both the horizontal and vertical components of velocity builds as the runner pushes harder and harder against the ground while gaining speed.
Hope this helps.
helps.

A: I'd like to make a comparison with pole vaulting, specifically pole vaulting with a rigid pole.
(A search with search terms such as 'evolution of pole vaulting' gave images illustrating pole vaulting with a rigid pole.)
In pole vaulting most of the height gain is from conversion of horizontal speed to height. (Some of the height gain is provided by muscle strength of the arms.)
I get the impression that in high jump the jumping leg doesn't bend all that much. The leg muscles of the jumping leg do get to contribute some height, but not all that much, it seems.
I get the impression that to an extent the jumping leg is utilized as a pole.
I don't know how elastic human tendons are, but possibly some of the conversion goes via storing elastic energy in the knee tendons. (For comparison, kangaroos have a specialized achilles tendon that is elastic in just the right way. When jumping around the kangaroo's muscles do not need to contract, they only need to resist elongation, which costs much less energy than active contraction. The elasticity of the tendon allows the kangaroo to reuse energy, rather than expending energy at every jump.)
Much of the mechanics of the high jump is in the swing of the free leg. It seems to me: the more vigorous you can swing up that leg during takeoff the more height you can gain. Again this is conversion of horizontal speed to vertical speed. With the running approach the free leg has already some horizontal speed to begin with, thus providing more available initial horizontal velocity to be converted to height.
Still, the jumper cannot afford to convert all horizontal speed to height. Since the jumper clears the crossbar with his back turned to the bar his path over the bar is necessarily a very diagonal path. The vertical acceleration of the jumper's center of mass is a given: that's gravity. So for the highly diagonal path more horizontal velocity is needed than when clearing the crossbar at right angles. 
Conversely, when jumping from a standstill position some of the precious energy has to be expended on causing the horizontal velocity needed to clear the bar. Not as much horizontal velocity as with the diagonal path, but still.
A: A significant part of the 55cm difference between a standing high jump and a high jump where athletes need to clear a bar is because, when clearing a bar, athletes don't have to get their entire center of mass over the bar all at once, only portions of their body.
This is enabled by the Fosbury Flop, which was popularized by the gold medal performance of Dick Fosbury in the 1968 Summer Olympics.
During the Fosbury Flop, the athlete approaching the bar turns during their leap, so that their back is towards the bar. As their head and shoulders clear the bar, they continue the turn, so that while their abdomen is clearing the bar, their head, arms, and feet are all below the plane of the bar. Kicking the feet high at the end lets them finish their jump without knocking the bar loose.
In addition to the comparisons of converting horizontal inertia into vertical energy in the other answers, this allows high jumpers to clear a pole that is up to 20cm higher than their center of mass could normally allow.
A: Other things equal, the faster you are traveling when you leave the ground and can't push any more, the higher you'll go.
But by the rules of the sport, you also have to go sideways some. Much of the effort you put into going sideways while you are pushing up will be subtracted from the vertical effort.
So if you are already traveling sideways at the right speed when you push off, you don't have to put that effort into going faster sideways. 
As a first approximation, you want to reach your highest point when you are at the bar. The faster you are traveling upward when you leave the ground, the longer it will take you to reach your highest point. And so the higher you go, the less sideways velocity you need. But human physiology determines how fast you can leave the ground, and you might do better to run fast and leave the ground farther from the bar, even though the same vertical speed could let you run slower and leave the ground at a closer spot.
So there are complications we can't deal with by assuming a spherical high jumper.
A: You don't. Velocity is a vector, and the speed you are accounting is in the x-coord which is perpendicular from y-coord. To illustrate this think on the time it takes a bullet to touch the ground when shot (without air)... it will take the same amount of time as if you drop it without shooting, so it does't account for the y time lapse.
With ax=0, ay=-9.8 m/s^2 = g, vxo=vxo and vyo=0.
if vxo=0, then vy=gt , vx=0, y=yo+vyt+gayt^2/2 and x=xo
if vxo\=0, then vy=gt , vx=vxo , y=y0+vyt+gayt^2/2 and x=xo*t. which is the same result for y.
