The last ten centimetres
This site has already carried a full cup across a room and got it safely to the doorway. That entry stopped at the table, which is where most of the tea is actually lost. This one is drawn rather than written. It is the second drawing here and the first laid out against a clock, and the words under it are a transcript, not an argument. The heights in the bottom trace are schematic, not measured. The shapes are what the physics predicts, and the sizes are there to show them.
In words, for anyone the drawing does not reach. Three traces share one time axis, ten seconds long, with a dotted line at 8.6 seconds, where the walking stops. Feet: speed rises from nothing to walking pace over the first second, holds flat until 8.3 seconds, and is back to nothing by 8.7. Hand: the cup’s height is level for nine seconds, drops ten centimetres between 9.0 and 9.85 seconds, and stays down. Tea: how far the surface climbs the cup’s wall, drawn against a dashed line for the brim. It rises halfway to the brim as you set off, bumps higher once near 2.6 seconds (the fourth-step wobble of the earlier entry, let die), then settles to a low ripple. At the stop it goes over the brim from 8.5 to 8.9 seconds, which is the drawing’s only tall peak. It sinks while the cup is lowered, then crosses the brim a second time, briefly, at about 9.9 seconds, when the cup meets the table. A bracket under the axis marks the final second.
Nothing on the walk is as violent as the stop. Tea leans against whatever push the cup is given. Brake at a fifth of the force of gravity and the surface tilts by a fifth, which in a mug eight centimetres across lifts the far side by most of a centimetre. You set off gently, taking a whole second to reach speed, so the tea leans a little and comes level again. But nobody stops gently. A walk ends in one footfall, and most of the braking happens when the leading heel lands, which takes less time than one slosh of the tea. A push that quick does not just tilt the surface. It throws it, and liquid thrown from rest swings well past the angle it would have settled at, toward twice that. The brim you filled to, because the brim was the amount you wanted, had a few millimetres to spare.
Then the precautions come off, one after another. The earlier entry found the carrying elbow clamped to the ribs, so the arm could stop being a lever and become part of the body. To put the cup down you have to unclamp it and reach. The last ten centimetres are travelled on an outstretched arm, a lever again, at exactly the moment the tea is at its worst. And the eyes, which that entry sent ahead to the doorway, have nowhere ahead left to go. They drop to the cup and see the wave.
What follows is the correction the earlier entry warned against, this time with a deadline. The wave is visible, the table is right there, and the obvious move is to get the cup down before the tea goes over. So the lowering is hurried, and a hurried lowering ends in a small knock against the wood, which is the second spike in the drawing. Nine seconds of the hard part go perfectly and the tea is lost in the last one, by someone trying to be finished.
The fix is to stop a stride short and wait. Brake over the last two steps instead of the last one, the way a train comes into a platform. Then stand still for a second with the cup held out over the table, which feels absurd, and let the surface go flat before the cup goes down. Nobody does this, because by then the walk is over and the waiting looks like nothing.
You arrive when you stop walking. The tea arrives about a second later, and the spill is what happens when you put it down before it gets there.