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A way to calculate something about a ferromagnetic ball in a magnetic field? Answered

I've built something like this https://www.instructables.com/id/Mechanical-Rail-Accelerator---Non-electromagnet-%22r/ and now I am thinking of calculating the kinetic energy of a ball flying off the magnet if it was 'set off' by a ball at some velocity. I have a great lot of problems:
1 The ball is about the same size as the magnet, which makes it impossible to say it is 'infinitely small'.
2 It is near the pole, so the magnetic field is not even (not like inside a coil).
3 The ball is ferromagnetic, so the field is going to be even more disrupted.
4 There is no Physics teacher this week.

Maybe (I hope) there is a formula of potential energy of something like my ball in a magnetic field? Any ideas? 


Why not measure rather than calculate? Mass and speed will give you kinetic energy. Speed can be measured directly with appropriate equipment, or approximated (modulo air resistance) by determining the parabola the projectile is taking and back-calculating from that

Well, I tried to measure it as the maximum height it reaches if rolling upwards, and got some crazy (I think so) results, so I want to know exactly what the theory predicts.

That's actually a good way to do the measurement and calculation. As you noted, trying to do the "potential energy" calculation for a highly non-uniform magnetic field is going to be awful. You can do it, but you don't want to without access to some very good numerical intergration software and a decent field map.

For the "how high does it roll" calculation, you'll only be neglecting friction, so you will end up with a slighty lower answer than reality). If you think you did the calculation wrong, you could post it here. I, or Steve, or Ork*, would surely be happy to look it over.

You're doing some very nice scientific investigation here, and a good job at thinking about and accounting for complications. From a working physicist, thanks!

I was thinking that if I leave a ball in about 1 mm from a 'stage' (a magnet and, let's say 2 balls) a ball would fly off with some Ek, which equals Ep of the ball at the beginning, so if it then flies to the next stage it will have both kinetic and potential energy, so the third ball would fly off with Ek2=Ep+Ek1+2Ep, so the maximum height will be doubled, but in my measurements it is roughly the same (11-15 cm with 1-4 stages). Is was I was thinking wrong?

Thank you for your help!

Your thinking is basically correct, and I don't see anything wrong with that analysis. Each stage should be additive. However, there are always losses which may not be controllable, and may not be linear, either (e.g., saturation or "terminal velocity" type of effects). Definitely go with the direct measurement if you can.

How do you blokes hold your heads up with such giant brains? Asked with the most respect possible whilst trying to be mildly amusing.

I hold my head up because of my giant brain :-) Mmmmmm....braaaiiinss...

Neat project. You really pick some fun ones don't you ?

I'd just add if you use a hard, non-magnetic surface, you might be able to neglect friction, for a first order calculation, and maybe balls of different sizes might give you a handle on frictional effects too.