Just a programmer, though been interested in physics since I was a teen and did take a bachelor degree in simulation (mainly physics).
Long ago though so rusty, $dayjob doesn't involve any advanced math at all.
edit: To expand, the "rough spreadsheet integration" was just the Euler method[1] assuming a constant acceleration. So
v(t+dt) = v(t) + a * dt
The acceleration comes from F=ma as mentioned, where F is the force of the engines (Newtons), m is the mass of the rocket (kg) and a is the acceleration (m/s^2). Solving for a we get a = F/m and we get
v(t+dt) = v(t) + F/m(t) * dt
To make things easy I assumed the weight of the rocket was constant at each timestep, but if we take dt to be small enough it's a decent enough approximation. For each timestep I also updated the mass using the estimated mass flow:
m(t+dt) = m(t) - 650 * dt
I started with m(0) = 377000 kg, v(0) = 1250 km/h = 347 m/s, and a constant -31850000 N force from the engines.
Using dt = 0.1 seconds, I got almost exactly 4 seconds until the velocity reached zero.
Newton's laws of mechanics are high-school physics IIRC; my son studied them at 8th grade or so. They are really simple; an evening with Wikipedia or 3blue1brown or whatever floats your boat will let you get sufficient understanding, provided you're also comfortable with high-school math.
