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It doesn't need to be more complicated than movement efficiency + horsepower + structure as far as vertical jump goes.

You must define the type of jump first though... and this should be a whole other thread.

wow great response, my physics knowledge is elementary here.

So that's my question, how are the forces involved in  a loaded depth jump different than an unloaded depth jump when the GRFs are the same?

GRF 100kg athlete dropping from 1m GRF who takes 0.15s to decelerate from 4.4m/s =
440/0.15 + 981 = 3914N

So the exact mass the athlete would need to add to produce the same GRF from only a 0.5m drop:
v = 3.1m/s

rearranging grf= mv/t +ma
grf/(v/t +a) = m

you get 27.5kg the athlete would have to add. But my question is, would the time it takes for the athlete to decelerate be the same?

Then furthermore as you mentioned, the time it takes to produce force by the athlete would certainly be less, increasing ground contact time, but not necessarily deceleration time, the time used to measure the GRF.

Good question.  First I need to remind you of one caveat.  We are calculating average GRF.  During the deceleration than instantaneous GRF increases and peaks and decreases.  You can measure the maximum amplitude using a force plate (and maximum might be most important for injuries).  You can imagine cases where you have the same GRF but different curves (eg you fall with almost straight knees and then bend vs a smoother landing where you bend nicely on impact) BUT I would bet for the most part athletes land in the same manner most of the time and average GRF is a pretty good substitute for our cases (could be wrong though - especially when you considering loaded landings may change peak force even if deceleration time is the same). 

So that partially answers your first question... even if the time to decelerate is the same MAYBE peak GRF is higher.  However... is time for athlete to decelerate the same?   Good question.  I don't really know.  For a depth drop maybe; you could make the case that altitude landings could be substituted with lower weighted landings (this might not be true but I haven't seen proof to contrary).   However, for a depth jump I would would argue that the time for athlete to decelerate would not be the same.   I would argue that you should think about two extremes - one is the ultimate reactive athlete who has springs (those perfect undamped springs you learn about in engineering courses) on his shoes and the other is the athlete who is completely inelastic.   The completely inelastic athlete lands and does not store any of landing force then initiates a jump.  His jump is the same as his standing vertical jump.  His deceleration time will be the same whether you add weight or height but his acceleration time will be longer (and less) if he is wearing load.   Essentially his deceleration and acceleration time are independent.   The athlete with springs is exactly the opposite; his landing is a completely elastic collision.   His deceleration time will equal his acceleration time.   He lands and his springs compress and store the force of his landing and then they decompress and return the force so he goes right up to the height he fell from (remember - frictionless).

Now, obviously you are not either of those examples.  You are somewhere in the middle.  You don't just want to be able to return energy; you want to be able to create it.  But you also want to use elasticity as much as possible so your jump is forceCreated + forceReturned = great jump.    If you have ever been trampolining you should know the feeling.  If you try to jump really high on the trampoline the experts will laugh at you are tell you not to "muscle it".  But in reality they are muscling it... The trampoline doesn't create energy it just returns it.  So when they go much higher than you they are using muscle to create force.... they skill is just to maximize the forceReturned part of the equation while also adding some of your own...

Given that I think you can see that if the goal of depth jumps is to practice at improving the forceReturned part of the equation we don't want a heavy load added.  To maximize force returned deceleration == acceleration because it is a return of force.  In reality we are using tendons to store force AND muscle to add force (because we aren't truly elastic) but we want to train ourselves to be elastic as much as possible; so any addition of weight which makes our acceleration longer would not be beneficial.  Now.... You could argue that weight is fine as long as you spend equal time decelerating and acceleration you are still training an elastic return and the elastic structures will improve and you will get better as using them...... However, I would imagine the elastic structures of a human have a pretty short time that they can store force and if you are too far outside of this you won't get much training benefit. 

edit: then another question is, is the GRF produced even important to consider in training for jumps?
How do quick depth jumps versus maximal height depth jumps differ in their GRFs?
I would think that quick depth jumps that emphasize minimal ground contact time would produce greater GRFs upon landing than depth jumps form the same height involving jumping as high as possible, but then how would the muscle-tendon unit be trained differently form those two types of jumps? Would there be any benefit to using a speed depth jump when it results in shorter jump height?

This is really the million dollar question isn't it?  The answer is I don't really know.  Physics is rock solid.  I can tell you what's right and what's wrong.  But physiology really isn't.  Human vary and what works and why it works isn't always so obvious.  That's why broscience creeps into training methodology so often.    We do know that humans are elastic.   On a basic level the GRF produced is VERY important to consider in training because GRF is what causes injury!  But as far as whether or not it causes the muscle/tendon to adapt and make one jump higher?  I don't really know. 

The difference with quick depth jumps vs. maximal height depth jumps (assuming the drop is the same) is that quick depth jumps may have greater GRF than maximal height but max-height never have greater GRF than quick depth jumps.

I would argue that if we believe depth jumps are important than the is certainly a benefit to using a minimal GCT depth jump even if it results in shorter height.   Consider that shortest GCT depth jump you can imagine.  Drop off the box, keep the knees straight and land and jump.  GRF will be high (a lot will be absorbed by your knees - hopefully you don't get injured) but some force will be stored in the muscle/tendon of your ankle.  The elasticity of your tendon will be trained.   Think of it as isolation for your ankle elasticity.  If this is a weakness of an athlete then this is beneficial.   In general the goal is to get the athlete to LEARN to be reactive so his total jump height will be better when he adds reactivity and power.   To force the athlete to do this we give him a goal (jump high) but keep GCT short so he he has to rely more on reactivity than on creating power.   Now, whether this LEARNING is just neural coordination or actual adaptation to the tendon or not... I don't know.   Whether this can actually be learned to much of a degree (would that athlete not have already figured out how to do it from just trying to jump high without a depth drop?)... I don't know....  But the training of short GCT does isolate the elastic ability of the athlete. 

I think an important difference in between a short GCT (like a low box depth jump over a hurdle) vs. a maximal jump depth jump (like a low box depth jump with an overhead target) is the fact that the mechanics are utterly different from a point on.

The landing mechanics are the same so you might think that eccentrically you're loading the muscles and tendons in the same way in both of these scenarios.

But starting with the amortization phase there will be differences, and the MOST differences will be at the point of full body extension.

In the obstacle depth jump you will get your knees towards your chest in order to pass the hurdle in front of you so the actual hip extension might not fully occur. You might short-cut it in order to have enough time to hip flex your knees towards your chest.

In the overhead target depth jump you can truly focus on maximal triple extension.

Also, since one has a length parameter involved (jumping forward over an obstacle) and the other is 100% vertical (in a perfect world), then even the landings will differ and the body's position at landing will as well...

So it's more than just a philosophical difference that needs to be resolved.

Wow haven't updated in awhile. Just being lazy really, been a bit demotivated with no gym!

Ran last Saturday. Did the 100m, high jump and long jump. It was a bit of a mess. The high jump started about 40mins before the 100m. I scissored 1.31m pretty easily and was looking forward to getting up to 1.50-1.60m, but they called the 100m so I went over. The electric timing was stuffing up though so we ended up waiting about 40mins (!!) before we ran. I was in heat 4 and by the time we got on the blocks I was just not in a good state to run. Ended up running a crappy 12.3ht, and annoyingly I missed the rest of the high jump because the 100m took so long. Did one long jump but it was rubbish. So overall a pretty crap day. My tib-fib pain had been improving though, only hurt a bit during warmups.

Because I don't have a gym at the moment I did two sessions at home. In a bit of a weird spot with no regular gym for a little bit. My plan was to repeat the power block that got disrupted initially for 3 weeks then test but that's kind of gone out the window unfortunately. Oh well. Time to transition into more jumps and less gym anyway.

Gym x 2:

Dynamic warmup/foam roll etc.

SL squats: 2x5/leg@~5kgs, 2x5/leg@~15kgs

UB, core

15 minutes of HIIT on spin bike (this felt pretty hard, really got a good lactic burn going. Might be a good way to get overall fitness up).

Track:

Dynamic warmup

4x15m falling starts

100, 200, 300, 150, 100 (80-90%)

Pullups: 3x8

This was pretty good way to get my special endurance work going. I also talked to the club coaches who rocked up as I was finishing and they train Mondays and Thursdays. I think I will start training with them as soon as I move closer (two weeks). Probably do one sprint session and one jumps session (long jump and high jump). So next block will look something like:

Monday: track

Tues: gym

Wed: rest

Thurs:  track

Fri: rest

Sat/Sun: either compete or maybe do gym/jumps

I also went to the physio to follow up my tib-fib pain. Basically, it had stopped pinching at the side of my knee during sprints, but the joint itself still felt quite sore. He thought maybe it was just residual inflammation that wasn't clearing properly and put me on 25mg Voltaren for about 4 days. Seems to have done the trick, sprinted last night with no real issues and feel good today.

Here's me running on Sat (in tights). Decent start but I faded bad...

2.98m? That's insane

You should film them

Yeah that's crazy far.  I don't get the double leg bounds though... 36 feet??  If you can jump three meters on a broad jump, that close to 10 feet...  I usually jump closer to 9'6 but I always go well past 40 feet in four jumps; the bound should help you go farther and farther and you should also benefit from the fact that when you do consecutive jumps you jump from where you land rather than go back and put your toes where you heels land (as you would if you were summing single jumps).

Maybe your on to something with the low jumping:

https://www.youtube.com/watch?v=ImXLBmzCRXg

In this video I do a standing jump at 30 seconds and three consecutive jumps at 35 seconds.  You can see from the yard markers that I barely clear 9 feet (maybe 9'3) on the broad jump; but on the multi-jumps I start four feet behind the line and land between the 9 and 10 yard marker (so about 32 feet with each jump going a bit farther).   I think I go a bit higher than you; maybe I do that because it is easier for bounding to jump higher but a low jump might help for standing jump??

not to detract from the ~3m jumps but he's doing them into a sandpit, which is lower (a bit more airtime) and softer than jumping onto turf or grass. still far for sure but not directly comparable with a turf-turf jump.