I have a large steel gate that slides open and closed on a steel track
with rollers.  It takes a couple of guys to get the gate moving and
they let it slam shut on a steel stopper.  Eventially the gate broke the
stopper off.  I would like to know the impact force of the gate
hitting the stopper.

The Weight of the Gate is 1360 pounds
I assume the velocity of the gate before impact was 5 miles per hour
The impact seems to be more of a plastic collision than an elastic collision.

I considered F=M*dv/dt, but the time it takes for the impact to take
place (dt) seems instantanious.  If there is a physics equation for
this problem or test data that I can relate to my problem it would be
helpful.

Seems like a lot of trouble to get to the real question of, "What kind
of stopper should I use that will not be broken by the gate?"

You need to know the stiffness of the stop, and the shape, to
essentially determine a spring constant.

From there, you can think of the stop as basically a spring (F = k*x)
that slows down the gate.  As the gate slows, the deflection and force
on the stop increases.  The peak force is when the gate is at zero
velocity.

Then, you'd have to look at the material strength (which is also
cross-section dependent).

Try looking up cantilever beam equations for limit and ultimate loads
for the material you're going to use (steel probably).

Exactly how did the stopper break off? Was it welded on, and the weld
broke? Or did the steel itself actually crack?