Explaining the phsyics behind a coil gun I made a coil gun for my extended essay, but I'm having some trouble explaining exactly how it works. The research question is how does varying the initial position of the projectile affect the efficiency? This is done by comparing the electrical energy in the capacitor bank to the kinetic energy of the projectile. 
So far my explanation goes something like this: The force - displacement graph for a magnetic field at a fixed strength is shown in green below, though I can only qualitatively explain why; As the projectile is in the center, there is an equal amount of ferromagnetic material on either side of the coil, and nett force is zero, however as the projectile moves from the center, a differing amount of material is present, so a force is exerted towards the center, increasing to a maximum point where the entirety of the projectile is on one side of the coil. Further than that the field strength decays, eventually reaching zero. The yellow curve is the current against time graph across the coil. My physics teacher told me to multiply the two graphs, and the resultant graph (in blue) should have the force experienced by the projectile along the y axis, and time along the x, and as such the integral is the impulse

I'm a bit unsure of how that works, especially with regards to both equations and simply the units. How does (dF/ds) * (dI/dt) = dF/dt ? If someone could explain what equations should be used that would also be very helpful. To be honest any insight at all would be helpful as I'm kind of panicking haha

 A: Graph 1: 
Force x Distance is work.
Graph 2: 
Power=V x I, and V is presumably constant, thus
Current x time is proportional to work.
Consequently the multiple together of x-step and y values of the co-ordinates of a point on either graph is work. The total work done is the integral of this, or area under the curve.
This provides a mechanism to know the distance/time graph, as the direction of motion is known, when the same total amount of work is done on each graph the points correlate.
More importantly, if you difference the electrical and mechanical work, you will measure the energy loss, which is essentially efficiency. You need to do different measurements from different starting displacements.
Multiplying the graphs together may be some kind of shortcut, I haven't quite figured anything out at this time. Or you might have misinterpreted your tutor, I guess.
I don't know what level you are working at, but I think the exercise in refining the theoretical answer to the question, is something you should develop more of yourself; that is it doesn't seem like you are stuck, only that you have stopped. There are a number of influences you will need to consider, including retarding work as the projectile exits. 
(I guess if you get really stuck, you can use this device as a weapon to demand answers - lol.)
