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Few days ago I played with ball(filled with air) in swimming pool. I observed interesting phenomenon.

When I released a ball from 3 meters depth the ball barely jumped above the water surface but when I released it from 50 cm depth it shoot out of the water like nothing.

I observed when released from 3 meter depth the ball goes up in zig-zag trajectory but from 50 cm depth is goes in straight line.

I would be very interested in calculating optimal depth from which the ball would jump the highest above the water surface. And as well I would like to calculate trajectory of the ball under water.

It is obvious that simple drag formula won't help here.

I guess that the zig-zag patterns is happening because there might be something like Karman vortex street behind the ball.

So have anyone idea how to calculate this? Or can you point me to the right literature?

Edit: I forgot one observation I made. It seamed to me that the ball when released from 3m depth was rotating when it hit a surface and that might prevent the jump.

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Nice hypothesis! You even have a formula to test given in the ref you cite. However, you have omitted measurements of some critical observable variables (present in the formula) from the question. Did you try evaluating the formula using the f, V, D, and Re you can observe or estimate from the actual system? Can you eliminate your hypothesis in this way? How might a ball differ from a cylinder? – Mark Rovetta Jun 14 '13 at 23:32
    
which formula do you exactly mean? By any formula there I would get straight trajectory and that is definitely not going to predict my observations. One thing I forgot to mention. It seamed like the ball when released from 3m depth was rotating when it hit the surface that might stopped the jump. – Tom Jun 15 '13 at 9:16
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I am going swimming in an hour or so and will try your experiment. – anna v Jun 15 '13 at 13:30
    
Ball was deflated and I could not find the inflating thingamajig . My view is that when you put the ball down with enthusiasm you give it more potential energy ( including the elasticity of the ball) and it bounces out high. When you dive, rather a friend dives, to three meters the ball is released at rest except the potential from the buoyancy. My analysis is that going up the three meters the ball gets a limiting velocity ( viscosity) and this is what is seen which obviously in your case is less than the 50cm one. I would try to have the ball at rest at 50cm and then release it to measure. – anna v Jun 16 '13 at 3:47

At 50cm 'depth' the ball was not actually submerged, because the water did not have sufficient time to refill the 'crater' created by the ball entering the water. The ball can be said to be 'floating' on the base of the crater. As the water returns at very high speed to refill the crater the bottom of the crater with the ball 'floating' on it is effectively uplifted through the air in the crater (not water). Once the crater is refilled the ball is now at the surface level of the water; having traveled at high velocity from its original resting place at 50 cm depth. It now leaves the surface of the water with the momentum gained by its high velocity rise through the crater and flies into the air.. seanmiddleotn99@gmail.com

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Welcome to Physics SE. Look around, and take the tour. In the case of this answer, note that the OP was not dropping the ball in to the water, but releasing it from underneath the water - there is not 'crater' to refill. That kind of invalidates your answer. – Jon Custer Jan 20 at 15:46
    
Agreed Jon, apogs. I was speed reading and jumped to a wrong conclusion. I would be interested to understand the physics of the often YouTube'd video of what I describe though. Complex to model accurately I suspect. – sean Jan 20 at 16:01

Try to eliminate your initial hypothesis by using dimensional analysis to make a quantitative estimate based on variables you can observe and measure easily.

You should be able to estimate a Strouhal number for your system from your observations of ball diameter, 'zig-zag' frequency, and the time for the ball to reach the surface from a known depth (i.e. fluid velocity.) That should be near 0.2 for your Karman vortex hypothesis to be a viable hypothesis. If its not you may want to consider an alternative explanation.

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I'll give it a try. – Tom Jun 18 '13 at 20:56
    
Can't really say for certain whether this was caused by vortex-shedding, but the math doesn't rule this possibility out if the ball was 10-15cm in diameter and if you saw it make 4-6 zig-zag cycles as it rose from 300 cm depth. – Mark Rovetta Jun 18 '13 at 23:06

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