Illustrated Physics
physics, drawn
Concept

Terminal velocity — where it appears

The speed at which a falling body's drag balances its weight, so that it stops accelerating. It sets the scale of every drag problem: the ballistic parameter that decides a trajectory's shape is the square of the launch speed divided by this one, and a projectile launched far above it spends most of its flight being slowed.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A metre of tube, with and without the metal. The magnet's position against time over a 1.00 m drop, integrated from Newton's second law with the eddy-current drag included, and the same fall with the drag switched off. The braked descent takes 2.09 s against 0.45 s in free fall. After a transient of about 50 milliseconds — the mass divided by the drag coefficient, and the only timescale in the problem — the trace is a straight line, which is to say the magnet is falling at a constant 49.0 cm/s. Every joule of the 118 mJ of gravitational energy released has gone into resistive heating of the wall, and none into the magnet's kinetic energy, since that is the same at the bottom as it was a moment after the start.

The magnet that falls slowly

Drop a magnet down a copper pipe and it takes several seconds to fall a metre, drifting rather than falling. Nothing touches it, the copper is not magnetic, and there is no circuit anywhere. The pipe arrives warm.

electromagnetism · Induction The same launches with drag 2.4 times the weight. Five launches at the same speed and at 20, 32, 45, 60, 70 degrees, drawn twice: in vacuum, where the arcs are symmetric parabolas, and with quadratic drag whose force at launch is 2.4 times the projectile's weight. Nothing about the drag figure is a parabola. Each path rises at nearly the vacuum angle, loses horizontal speed that nothing restores, and comes down far more steeply than it went up — the 32° launch leaves at 32° and arrives at 50°. The best of these angles in vacuum is 45° and in air is 32°, and the best range has fallen by 60 per cent. The asymmetry is the whole of the difference: drag removes speed in proportion to speed squared, so it takes most from the fast early part of the flight, and the descent happens at a speed the drag has already limited.

The angle that drag moves

Forty-five degrees is the answer in vacuum and almost nowhere else. Add one velocity-dependent force and the two equations of motion lock together, the closed form disappears, and the best launch angle falls — to thirty-eight degrees for a golf ball's drag and to twenty-nine for a shuttlecock. What moves it is not the loss but the asymmetry.

mechanics · Projectile A drop that sweeps up mist settles at a seventh of g. The acceleration of a water drop falling from rest through a still mist that it sweeps up completely, in units of g, against time, for drops starting at radii of 0.2 mm, 0.05 mm and effectively zero, with no air resistance. The mist holds 10 g of water per cubic metre, twenty times a typical cloud, so that the approach takes seconds rather than minutes. Every drop starts at g, because at first it has gathered nothing, and every one settles to g/7 = 0.1429 g: after 2 s the three are at 0.377, 0.172, 0.143 g, and after 8 s at 0.144, 0.143, 0.143 g. The drop that starts from nothing is at g/7 from the start. The final acceleration contains no property of the mist, the drop or the water: a denser mist makes the drop grow faster and reach g/7 sooner, and nothing else.

The raindrop that falls at a seventh of g

A drop of water falling through a still mist, gathering every droplet in its path and with no air to slow it, does not fall at g. It settles to an acceleration of exactly a seventh of g, whatever the mist, whatever the drop, and its radius grows by the same amount for every metre it falls. Of the work gravity does on it, three-sevenths goes into the gathering and never comes back. The same reasoning gives a third of g for a chain paid out from a heap, and a single rule — g/(2n + 1) — for anything that sweeps up mass as the n-th power of the distance it falls.

mechanics · Momentum

Named alongside it

The objects these essays reach for when they reach for this one.

DissipationBallistic coefficientCloud microphysicsConductivityConservation of energyDampingDragEddy currentsElectromagnetic inductionFlux freezingInelastic collisionJoule heating

All concepts