A second coil probably is a good way to detect the passing of the projectile. If you have two coils wrapped around the barrel like so:
coil 1 coil 2
==//////|===/////|===
| | | |
+ GND A B
A voltage is applied across coil 1 to accelerate the projectile. This acceleration is caused by a change in the magnetic field inside that coil. Let's say it is in the rightward direction. If this is to accelerate the projectile to the right, the projectile's magnetic field must be directed left. As this projectile approaches coil 2 from the left, it causes an increasing, leftward magnetic field in coil 2. This, in turn, induces a voltage in coil 2 that will oppose the changing magnetic field. So there will be a current in coil 2 that creates a rightward magnetic field. Depending on which way you wrap the coil, you will get a positive or negative voltage across coil 2. As the projectile gets farther away, you will see the opposite voltage induced, as the magnetic field of the projectile is still directed to the left, but is decreasing in magnitude. This can be measured with a voltmeter (attach the leads to A and B) just to see the effect. If you want to use that voltage to switch the first coil off, you will need to create a circuit or use a microcontroller such as an Arduino.
Long answer short, in the above configuration, you will see a voltage induced in the secondary coil as the projectile approaches, and the opposite voltage as it gets farther away.
a) not safe to plug into mains
b) need more copper wire, fill the available space with turns
c) decide how long you want it to remain attracting for, then aim to heat up your windings in that time, ie use as much input power as you can without overheating. Depending on the cross section of your copper, this may be a car battery, or several in series.
This will allow you to maximise the attraction for that core geometry. It may still not be enough for your spoon.
A steel spoon will be strongly attracted to a magnet. A stainless steel spoon, depending on the precise grade, will be somewhere between very weakly attracted to completely not attracted at all. Make sure you are using the right type of spoon!
Best Answer
What a wonderful time to be able to teach SAFETY, not only to your 10 year old, but to all of his classmates. Safety is always on topic here on EE SE.
Children are enamoured with magnetism, and displays of fireworks, and squished things.
Think this through VERY carefully. Consider the implications of a magnetism demonstration that builds upon safety.
And of course with the admonition of "ONLY UPON ADULT SUPERVISION".
Use the other comments and answers to guide you.
Then demonstrate how a grape can be squished (like a human's finger).
No worry's about radiation