Precision control of current to an inductive coil (coilgun)
The real problem with coil guns is not in turning on the magnetic field, but in turning it off.  That is therefore what you should be concentrating on and designing around.

The coil looks like a inductor to the circuit.  When you first apply a voltage to the coil, the current, and therefore the magnetic field, build up linearly in time.  The projectile starts out stationery, then accelerates as the magnetic field gets strong enough to overcome gravity and static friction.  That takes time.  You therefore have a relatively "long" time to get the field built up initially.

The reverse is true when shutting down the field.  Ideally, you want the field to be strong until the projectile passes the point where the field is pulling it back instead of propelling it forward.  This is the point when the projectile is moving fastest, so there is a much shorter time window to get it right.

Unfortunately, the inductance of the coil makes it impossible to shut off the current instantaneously.  The rate at which current in a coil decreases is proportional to the reverse voltage applied.  To stop the current instantly would require infinite voltage.  The speed of the transistor, or whatever you use as a switch, isn't so much the issue as the reverse voltage it needs to withstand.

Since you are driving the coil with 160 V, whether the switch drops a volt or two doesn't matter much.  You can use either MOSFETs or BJTs.  I would seriously consider BJTs in a real production design, but considering your circumstances I'd go with MOSFETs.  One reason is that they parallel better without your students having to get into some details that may be over their heads.

Use N channel power MOSFETs in parallel as a large low side switch.  Drive each with a separate off the shelf gate driver.  You'll need a 12 V or so supply for those.  The inputs of the gate drivers can be tied together, which gives you a single logic signal to turn the switch on and off.

So far you have the 160 V supply, the coil, and the FETs acting as a single switch all in series.  That works for turning on the coil, but you have a big problem turning it off.  The FETs can switch quite fast, but when they do, the coil will produce a large inverse voltage.  If you don't do anything about that, the voltage will go as large as it takes to force current thru the FETs in the short term anyway, which will damage the FETs.

You need to give this coil kickback current a place to go, while still letting the voltage go as high as you can handle, but not higher.  The higher you can let the reverse voltage go, the faster the coil current will stop, which means the longer you can keep it on without pulling the projectile back at the end.

400 V FETs are reasonably available, so I'd plan the kickback voltage to be a little less than that.  Let's say we aim for 350 V to leave some margin.  Since the coil starts at 160 V, that means the reverse voltage on the coil can only be 190 V, so let's say 200 V.

One way to achieve that is to put a diode with resistor in series across the coil.  The diode is oriented so that it conducts when the kickback voltage is present.  The resistor is sized so that 200 V is across it when the peak coil current is put thru it.

There are many other details to discuss.  But, this is getting long already so I'll quit here.