# Determining Vertical Jump Height from Average Velocity

1. May 12, 2008

### tom122079

I am a coach for athletes in various sports and have a problem that I can't figure out, but I'm hoping will be easy for someone here.

I am looking to determine the height of a vertical jump from some limited information via an accelerometer.

The accelerometer is attached near the center of mass of the subject. The subject then jumps as high as possible and the accelerometer produces the average velocity of the jump.

The average velocity is the average of the entire vertical movement of the center of mass. So from the beginning of the upward movement (feet still on the floor) to the peak height of the jump. To give you an idea of the type of velocities we are working with, it is typically around 1.50 meters per second.

I am looking to determine the height of the jump (or at least an accurate approximation) from this information. Is this possible?

My biomechanics and physics classes were so long ago that I can't seem to figure this out. Any help is greatly appreciated.

2. May 12, 2008

### Redbelly98

Staff Emeritus
[Edit added after posting: WAIT, does the average include part of the time the person is on the floor? That is, the accelerometer starts when the person begins upward movement, and not when the person is just leaving the floor?
If it starts just as the person is leaving the floor, my formula will work. Otherwise, I'm not so sure ... ]

Jump ahead to the end, if you want to skip the derivation

What we know is:

Average velocity (=Vavg)
Final velocity (=0, at height of jump)
Acceleration (=g=9.8 m/s^2)

From this we can get the starting velocity, as the person is just leaving the floor:

Vstart = 2 x Vavg
so that the average of Vstart and 0 (at the height) is Vavg.

One of the formulas for this type of thing is
Vstart^2 = 2 g h
so that

$$h=\frac{V_{\mbox{start}}\ ^2}{2g} = \frac{4*V_{\mbox{avg}}\ ^2}{2 g}$$

or

$$h= \frac{V_{\mbox{avg}}\ ^2}{4.9}$$

where h is in meters and Vavg is in meters/second.

Sanity check: if Vavg is 1.5 m/s, then we get h = 0.46 m or 1.5 ft. Sounds reasonable.

Last edited: May 12, 2008
3. May 13, 2008

### tom122079

Yes, the accelerometer starts when the person is on the floor as soon as they begin their upward movement. With this is mind is it necessary to come up with some approximation of the initial instantaneous velocity upon leaving the floor?

Is there any reliable way of doing this with the information we have?

Also, the accelerometer lets me input the weight of the subject and then gives me the average power output in watts. I would assume I could derive average acceleration from this data, but I am not sure this would help at all since this would also include the time spent on the floor.

4. May 13, 2008

### Redbelly98

Staff Emeritus
I am not familiar with the mechanics of the "pre-launch" part of the jump, i.e. the upward movement while the feet are still touching the ground, or how to express that motion in terms of useful equations.

If the accelerometer readout could be reprogrammed to display maximum velocity, this would be easier. I would think that somehow that information is in the processor when it calculates the average, and maybe that is or is not accessible to the user. Or maybe I don't understand fully how the accelerometer works. Do you have the manufacturer and model number of the unit (so I could do a Google search on it), or better yet a link to a website that explains how the device works?

Regards,

Mark

5. May 13, 2008

### tom122079

Here is the link to the product:
tendosport.com/index.php?file=fitrodyne&checkproducts=do
(you'll have to add in the "www" since I am not allowed to post url's due to my small number of posts on this board)

You can also check out the manual there.

To my knowledge there is no way to get the max velocity. The only information it gives is the average power and velocity.

I am interested to hear your thoughts.

6. May 13, 2008

### Redbelly98

Staff Emeritus
Tom,

The manual for the computer software,
http://www.tendosport.com/manuals/manual-wa-computer.pdf? [Broken]
says that graphs of displacement vs. velocity are displayed, and can be viewed for each rep or jump. I've attached an example graph from the manual.

Looks like you could just read off the maximum displacement to get the height of the jump. But we need to figure out the displacement reading when the jumper leaves the floor, since "zero" in the graph is when upward movement begins.

Can you post an example graph for an actual jump? It might look a lot different than the weightlifter graph in the manual. It would be good to see an actual jumper's graph.

Regards,

Mark

p.s. feel free to ask in here, or message me, if you have questions about how to post the graph.

#### Attached Files:

• ###### Accelerometer_displ-vs-velocity.gif
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Last edited by a moderator: May 3, 2017
7. May 13, 2008

### tom122079

That's a newer model than the one I have. I have the Tendo Powerlyzer which only gives the average velocity and average power. Unfortunately it does not give me a graph.

Any ideas?

8. May 13, 2008

### Redbelly98

Staff Emeritus
Running out of them, but will keep thinking about it some more.

9. May 17, 2008

### Redbelly98

Staff Emeritus
I've spent some more time thinking about this. It is not an easy problem, but I have some ideas that may or may not pan out.

For this discussion, I'm referring to 2 phases of the jump:
(1) the "pre-launch" phase, where the jumper is moving upward but still has his/her feet planted on the ground, and
(2) the airborn phase, where the jumper has left the ground.

The tricky (and unknown to me) part is: what is the force exerted by the jumper during the pre-launch phase? I doubt that it is simply a constant, as I imagine little force is exerted as the knees approach a straight (i.e no bend) position at the take-off point. The nature of this for might have been covered in your old biomechanics classes, but I do not know any simple way to derive it based on standard physics equations.

After looking at the "advanced model" manual which displays graphs of displacement vs. velocity, it seems to consider the jump (or lifting of the weight, its intended application) to occur only while the accelerometer is moving upward. So at the height of the jump, the accelerometer senses that upward motion has reversed and stops taking data. This makes sense for weightlifters, as you'd want to know the average velocity and power during the lifting of the weight. (It looks like you said as much in your first post, so that jives with what I saw in the manual.)

So here are two ideas to try out.

Idea #1:
Can you attach the accelerometer to a jumper's ankle? That way, the data used by the processor will only take place during the airborne phase of the jump, when the feet have begun travelling upward.
A possible problem with this is that a jumper could "cheat" by bending his knees at the height of the jump, artificially recording a higher jump height. But if you can either watch a jumper's legs carefully, or you are confident that an "honor system" works with your group, this could work.
In this case, the height will be given by my earlier formula, since only the airborn phase is recorded by the accelerometer:

$$h= \frac{V_{\mbox{avg}}\ ^2}{4.9 m/s^2}$$

(V_avg in m/s will give height h in meters)

Idea #2.
This would be experimental in nature. Can you measure vertical leap by other means, eg. a height scale that the jumper touches at the height of the jump? By recording the average velocity and actual jump measurement, perhaps a best-fit curve could be estimated to correlate the two measures. I'd suggest using jumpers covering a range of jumping ability, and maybe also have each make 3 jumps: a max effort, 1/2-effort, and 3/4-effort jump.
I'd be willing to help with the data analysis, if you'd like.

I think these 2 approaches are your best bet at this point, unless others in here have any other ideas.

Regards,

Mark

10. May 19, 2008

### tom122079

Those are both good ideas. I am going to try out Idea #1 today. It's so simple that I think it should work. It will still take into account a small amount of the time that the athlete is on the ground as the ankle extends, but should be much more accurate than attaching the accelerometer at the hip.

Idea #2 is good in theory. The problem is that a height scale as you described typically measures the difference between the highest point an athlete can reach while standing and the highest point an athlete can reach when jumping. This set-up is very easy to cheat (e.g. not reaching as high as possible on the standing reach), and errors often result even when the athlete is trying to be honest. There are some different ways to measure jump height that are more accurate than this, but unfortunately they all require the purchase of additional equipment, which isn't in the budget at this point.

My hope is that with the use of the accelerometer and Idea #1 the results should be very consistent and accurate.

I will report back with my results.

11. May 19, 2008

### robertm

If you want to avoid the slight upward movement of the heel lifting off the floor, you could attach the device to the toe of the jumpers shoe. Being clear to your students that they need to keep their ankles pointed straight down throughout the jump (I'm picturing a vertical jump from a standstill).

Also I would like to complement redbelly for his very elegant solution to this problem.

12. May 20, 2008

### tom122079

Well, I tried attaching the accelerometer to the ankle and the results were promising.

The average velocities were much less than those recorded with the accelerometer attached at the hip. Over the course of 2 weeks the athletes performed jumps with weights in their hands. The mechanics of this movement only differ from a traditional vertical jump in that the arms cannot be used to generate momentum and the velocities are slower to due to the additional weight.

Using one athlete as an example, let's call him athlete A, the results were as follows:

Week 1 (accelerometer attached at the hip): mean velocity= 1.45 m/s
Week 2 (accelerometer attached at the ankle): mean velocity= 1.13 m/s

Although the athletes were initially discouraged by the lower numbers, I explained to them that this was to be expected since we were taking the "pre-launch" portion out of the equation.

I know that I initially suggested that the pre-launch portion would not be completely factored-out due to ankle joint extension while still on the ground, but I was able to eliminate this problem with a feature of the accelerometer which allows me to set a "buffer zone". I set the buffer zone at 5 cm, which means that the accelerometer did not start recording data until it had been displaced by 5 cm.

I know that you may be thinking, "Why not attach the accelerometer to the hip and use this 'buffer zone' feature to factor-out the pre-launch portion of the jump." Well, the reason is that each individual athlete will squat to a different depth during the pre-launch portion of the jump. So I would have to determine the individual buffer zone for each athlete, and even then there would be variation within the individual.

By attaching at the ankle we eliminate this issue since the displacement of the accelerometer attachment is always right around 5 cm (it's not perfect, but it's pretty close).

Due to this factor I believe this is the optimal set-up. However, I did run into some inconsistencies in the data.

Using Athlete A as an example again, the data we collected in Week 1 (with the accelerometer attached at the hip) was very neat and free of outliers. Nearly every jump was between 1.40 and 1.50 m/s, and even those that weren't were pretty close.

In contrast, the data from Week 2 (with the accelerometer attached at the ankle) was much more variable. We had velocities as low as 0.96 m/s and as high as 1.46 m/s. When you plug these numbers into the formula you get 7.40 inches and 17.13 inches. Now obviously these are the extreme outliers, but there were several other data points around them.

Simply put, there is no way that an athlete would vary this much in jump height, provided he was giving maximal effort on every jump (which he was). 1-2 inches is the most variance I would expect.

I attribute this variation to horizontal movement. Although the athletes are instructed to jump vertically, there will always be some horizontal movement, which will displace the accelerometer and potentially increase the average velocity. However I am unsure of why this would be more of an issue with the accelerometer attached at the ankle instead of the hip. If anyone has other ideas on what may be causing this variation I would be very interested.

I will have to find some way to correct this issue in order to get truly accurate data, but I believe that the attachment at the ankle is the best way to go.

Mark, I truly appreciate your help. Usually the simple solutions are the best, and that is certainly the case here. Thanks again.

13. May 20, 2008

### rcgldr

If the goal is to measure vertical leap, then why not do it the old fashioned way, with a yardstick (or a board with a numbered line every inch) mounted on a wall or post? The person reaches up and touches the board, then jumps up and touches the board again, and the difference is noted.

14. May 20, 2008

### Redbelly98

Staff Emeritus
Hi Tom,

The 5 cm (2") buffer zone is a great feature.

Well, the large variability is surprising. I am trying to think of possible reasons, and possible ways to test those reasons. Here is what I can think of so far:

1. Horizontal movement. This can be tested by having an athlete do a horizontal jump with little vertical lift. A running jump might work, as the accelerometer will resets itself after every step. So if such a jump gives a very large number, horizontal movement is a factor. If the average velocity is low or close to normal, then it is only registering vertical movement and we can rule out horizontal motion.

2. Knees bending near height of jump. You might need to watch an athlete carefully during a jump to see if this is happening. If this is a factor, it means that the lowest reading (with knees kept straightest) is the truest one. Of course, keeping knees straight for the whole jump is definitely not advisable as injury could result. Don't know what to say except watch the jump carefully if you can -- I imagine things happen pretty quickly so I don't know how easy that will be to do.

Finally, it really would be good to comparison test with another height measurement if possible, like the one Jeff Reid mentioned. Perhaps you could try it yourself, doing different jump heights on your own. We'll assume for the sake of argument that you are trustworthy!

-- Mark

p.s. is a 7.4" jump at all believable? Are the athletes young children? Just curious.

15. May 20, 2008

### Redbelly98

Staff Emeritus
Thought of something else ...

3. Is it possible the accelerometer sometimes gets kicked by the other foot during a jump? Or can otherwise twist around unpredictably?

Another question, is it the unit pictured on p. 8 here:?
http://www.tendosport.com/manuals/manual-fitrodyne.pdf [Broken]

Last edited by a moderator: May 3, 2017
16. May 20, 2008

### tom122079

There are a couple of issues with this method. First, as mentioned before it is very easy to cheat the test, and even when not attempting to cheat the results can be inconsistent.
Second, we do a lot of "weighted jump squats" which are basically a vertical jump with additional weight held in the hands or with a bar across the back. Both of these methods preclude the use of the test you describe since the athlete cannot easily reach at the top of their jump.

My hope is that with the use of the accelerometer we can create a vertical jump test that is more accurate and harder to cheat, while also allowing us to get measurements during the exercises I described.

Funny you should mention this, because we also did some standing long jumps with the accelerometer attached at the ankle. For the athlete used in the example the velocity was around 2.20 m/s. This velocity was for a jump where no additional weight was used and the athlete was instructed to jump for maximum distance. This was also done with the 5 cm buffer, so most of the pre-launch phase was recorded.

In regards to this, it should be noted that the accelerometer is set-up slightly off to the side of the athlete's foot so the string from the ankle to the accelerometer is at a slight angle. I'm not sure if this matters or not.

I don't believe this is an issue. It would be pretty easy to see anything besides a very small movement. Furthermore, the athletes should keep their legs straight for the entire
duration of the airborne phase. The knees only bend as the feet contact the ground in order to cushion the landing.

Comparing between the tests could be done, but I'm not sure how much it would tell us. When athletes perform these jump tests and are simultaneously filmed on a 3D motion capture system the results are different. The 3D motion capture is the most accurate test there is as it measures the displacement of the center of gravity. It is not uncommon to see an athlete jump 30" on one of the reach tests, but this jump would likely be anywhere between 25-28" on the motion capture system. However, the results of a comparison may still be useful.

And no, the athletes I am testing this on are varsity high school football players. So, yes, 7.4" seems low. Although we need to keep in mind that these jumps were done with additional weight (about 10% of bodyweight) and without the use of the arms. Combine that with the fact that this test may yield a more accurate result than the typical reach test and it may explain the low number. However, by my estimation it still doesn't add-up. I would estimate the athlete in the example would have a 25" vertical if using a reach test.

The accelerometer is not being touched by anything during the test, so this should be a non-issue. It is also on a tension reel, so the twisting shouldn't be an issue either.

Yes, it is the same basic set-up, but the microcomputer screen is different because it is an older model. Same basic thing though.

17. May 22, 2008

### Redbelly98

Staff Emeritus
Thanks for the thorough responses! After reading that, and referring back to the manual, I now have a better idea how this thing works. What I didn't realize before is that the athlete/jumper is actually pulling a string or cable that is attached to the stationary sensor unit, and it's the movement of the string coming out of that unit which gets recorded and analyzed. Horizontal, vertical, it doesn't matter -- if the string is being pulled, it is registered as motion.

So here's another list of thoughts and suggestions, from me to you!

1. Could the batteries be low? I've noticed when using some electronic equipment that low batteries can cause flaky readings. Ideally the unit would have some "low battery" indicator, but that may not be the case.

2. You said
This could be the key. Having the unit off to the side means the string is not pulled as quickly when the jumper just leaves the floor, since it's pulled at a right angle and not directly away from the sensor unit. But this moment of take-off is just when velocity is at its maximum! The result is readings will be lower than the "true" average velocity, and the inconsistencies you saw could be due to slightly different placement of the sensor unit.

3. Here is something new to try. Can you attach a clip or small clamp of some sort to the cable, to prevent it from retracting into the sensor unit? Then you can attach the cable to the jumper's waist as you did before. With the jumper standing up, attach the clamp right where the cable comes out of the sensor unit, so that when the jumper squats down the line does not retract into the unit. The recording will start when the jumper's legs are straight again, that is at the end of the pre-launch part of the jump. The angle of the cable will not matter as much as when it was attached to the jumper at floor-level.

I hope I explained that clearly enough. The clamp would have to be such that it does not permanently put a kink in the cable.

Best regards,

Mark

Last edited: May 22, 2008