Is it better to "hit it firm" or to "play the line"?

You are told to "get it past the hole", and also to "use the break with a dying putt". Which is right? Looking at the physics of the cup itself, the answer is clear.

The cup rejects a ball that is going too fast

For the ball to drop, it has to fall by one ball radius between crossing the near edge of the cup and reaching the far edge. From that condition, the critical speed for a ball running across the middle is about 1.63 m/s. Any faster and the ball simply flies over the hole.

And the further the ball passes from the centre of the cup, the shorter the distance it has to cross — so the speed it is allowed to have drops too:

v_max(d) = 2√(R² − d²) · √(g / 2r)

In other words, the firmer you hit it, the narrower your margin for direction becomes.

Too soft and you cannot read the break

Go the other way and the ball breaks sharply at the end. Curvature is inversely proportional to the square of the speed, so in that last metre a small error in your read becomes fatal. And of course, a putt that never reaches the hole goes in 0% of the time.

The optimum is "just a little past"

The tolerance in pace and the tolerance in direction move in opposite directions, so there is an optimum somewhere in between. Once you add launch dispersion to the calculation, in most conditions the touch that maximises your chance of holing out is one that would finish roughly 30–50 cm past the hole.

That does shift with slope and green speed, though. On a steep downhill there is almost no room to run it past, so the optimum moves towards a dying putt.

Look at the "map of what goes in" in the Putt Break Calculator and you can see the shape of those tolerances at a glance. Tall and narrow means a line that forgives pace; wide and flat means a line that forgives direction.