Linear Induction Coilgun with superconducting projectile

Assuming this is a made up idealized problem and we're not trying to find new and wonderful ways of hurting each other, the super conductor should expel any applied field, no field no heat. If it is a real question then the superconductor would rapidly quench in the applied field. The resulting fireball would expand rapidly so be sure to duck.

 

The OP does not say if the gun is in air or in a vacuum. If air, the superconductor needs very good insulation to avoid warming by the friction heated air.

 

Sorry, I didn't realize this was a test. The reply I gave is still correct.

 

The surface of a superconductor is a perfect reflector of magnetic fields. A current will flow on the surface of the superconductor that generates the exactly equal and opposite magnetic field needed to cancel the incident field. So you cannot use a magnetic field to induce a current inside a superconductor, only a current on the surface.
There will be physical forces between the magnet conductor and superconductor resulting from the magnetic field. If movement occurs then electrical energy will be converted to kinetic or potential energy, but not to heat in the superconductor.
Since no energy enters the superconductor it will not be heated. If a superconductor was heated it would become a resistor and the scenario would change instantly.