Many people believe that a run-up helps them to jump higher.
Is this true?
(While it seems natural to do this, I'm having trouble believing that horizontal speed can help one jump vertically.)
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It is true.
See Young, Warren, G. Wilson, and C. Byrne. "Relationship between strength qualities and performance in standing and run-up vertical jumps." The Journal of sports medicine and physical fitness 39 (2000): 285-93.
Subjects were 29 males aged 19 to 34 with at least one year experience in a sport requiring sprinting or jumping.
The reason why is reactive strength. (Ball, Nick B., and Sara Zanetti. "Relationship between reactive strength variables in horizontal and vertical drop jumps." The Journal of Strength & Conditioning Research 26.5 (2012): 1407-1412.)
Reactive strength is an athlete's ability to use the "stretch-shortening cycle (SSC) to increase subsequent force production".
A drop jump is a common way to display and assess reactive strength.
Drop jumping is a plyometric activity that involves stepping from a predetermined height, landing, and immediately performing a maximum jump.
A runup similarly allows a jumper to make use of the stretch shortening cycle to increase subsequent force production.
From (Young et al. 2000):
reactive strength is relatively more important for jumping from a run-up than for the standing VJ
Also, not everyone is able to make effective use of the stretch shortening cycle — some peoples' drop jumps and runup jumps are similar in height to their standing jumps.
Yes. While there would be no vertical component to pure a pure horizontal velocity vector, direction can be changed via collision with the ground. To some degree it's like banking a billiard ball off of the rail and changing it's direction by 90deg. In the case of a running vertical jump, you were able to change the shape of "the rail" on the fly via your foot and ankle because gravity + static friction allow your foot and ankle to act as part of the ground system momentarily.
Separate from the horizontal component of the run-up, a pre-jump also allows a jumper to store energy in the elastic portion of their muscle/ligament/tendon system in a way like a very stiff trampoline.