Illustrated Physics
physics, drawn
Concept

Projectile — where it appears

A body moving under gravity alone after being launched, which follows a parabola when air resistance is negligible. Its motion separates into steady horizontal travel and uniformly accelerated vertical travel, which is why its range and height obey simple rules.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The boundary of everywhere a given speed can reach. Trajectories at one speed and five launch angles, with the curve that bounds all of them. The boundary is not one of the trajectories and is not the 45° launch: it is the envelope of the whole family, the locus of points where two neighbouring launches cross. Distances are in units of v²/g, so the greatest range is one and the greatest height a half, and the envelope is the parabola y = ½ − x²/2 joining them. Two things about it are worth having. Its focus is the launch point exactly — every point on it is as far from the gun as from the line y = v²/g — so the safety parabola is a conic with the same focus as the trajectories themselves. And a boundary made of crossings of neighbouring members of a family is a caustic: the same construction that makes the rainbow's edge bright, drawn here with cannon shells instead of light rays.

Everywhere a throw can reach

Fix the speed and let the angle be anything. The trajectories fill a region, and the region has an edge — a curve that is not one of the trajectories, that touches each of them exactly once, and that turns out to be the same kind of object as the bright rim of a rainbow.

mechanics · Projectile Every slope answered by one curve. The boundary of everywhere one throwing speed can reach, drawn about the hand, with straight lines from the hand at −30°, 0°, 20°, 45° running out to it. Distances are in units of v²/g. Because the boundary is a parabola with its focus at the hand, the distance to it along any direction is r = (v²/g)/(1 + sin α), and each drawn length was found separately — by searching every launch angle for the one that lands farthest along that line — and agrees with the formula to ten decimal places. At −30° the greatest reach is 2.000 v²/g, launched at 30.0°; at 0° the greatest reach is 1.000 v²/g, launched at 45.0°; at 20° the greatest reach is 0.745 v²/g, launched at 55.0°; at 45° the greatest reach is 0.586 v²/g, launched at 67.5°. Uphill the reach shrinks and downhill it grows without limit as the line approaches straight down, and the launch that achieves it always bisects the angle between the line and the vertical. The small dots are the foci of those best throws: every trajectory's focus lies on a circle of radius v²/2g about the hand, and the farthest throw along a line is the one whose focus lies on that line.

One curve answers every slope

A throw up a hillside, down one, off a height and into a basket look like four problems with four answers. They are one problem. The edge of everywhere a throw can reach is a parabola with its focus at the hand, and written about that focus it gives the farthest reach in any direction in one line — along with the reason the shot that needs the least effort is the one whose aim matters least.

mechanics · Projectile Range as a map of launch velocities. The plane of launch velocities — horizontal component across, vertical up — with the curves of equal range drawn on it. Launched and landing at one height, the range is 2vₓvᵧ/g, so every curve of equal range is a hyperbola vₓvᵧ = constant, drawn here at ranges of 0.25, 0.50, 0.75, 1.00 v²/g. A thrower who can produce one speed in any direction can reach any point on the half-circle of radius v, and the best throw is where that circle touches the highest hyperbola it meets — at 45°, where the hyperbola vₓvᵧ = ½ is tangent to it, because a circle centred on the origin is symmetric about the diagonal and so is the hyperbola. The famous angle is a property of the shape of the set of throws.

The best throw is a tangency

Shot putters release at about 37°, long jumpers take off at about 20°, a ball thrown forward from a moving truck should be aimed steeply and flies flat, and a golf ball's drag alone moves its best angle to 38°. Each is usually explained as an exception to 45°. None of them is. Drawn as a map over launch velocities, range has curves of equal value, a thrower has a set of throws they can make, and the best throw is always where the set first touches a curve.

mechanics · Projectile The launches that go in, and one thrower's scatter over them. Every free throw as a point: launch angle across, launch speed up. The dark curve is the launches that put the ball's centre through the centre of the hoop, lowest at the least-speed launch, 51.4° and 7.17 m/s. The shaded band is every launch that passes cleanly through, found at each angle by moving the speed until the ball touches the rim. It does not exist below 46.9°, is a hair thick near the bottom of the curve, and thickens as the launches steepen. The two ellipses are one thrower who scatters ±0.05 m/s in speed and ±1° in angle, drawn at two standard deviations and centred on two aims: the least-speed launch, and 58.5°, the aim that makes a clean pass most likely for that thrower. At the first, the ellipse lies across a band far thinner than itself; at the second, more of it lies inside, although the band there slopes more steeply.

The throw most likely to go in

A free throw can be launched at 51.4° with less speed than at any other angle, and there a small error of angle hardly moves the ball at all. It is still not the best aim. Once the question is which throw most often goes in rather than which is cheapest, the thrower's scatter has to be laid over the launches that succeed — and for a hoop the answer moves steeper, while for a board the same scatter moves it flatter.

mechanics · Projectile The drift is the area under the height. Height against time for the two balls: the throw (dashed curve) rising to 100 m in 4.52 s and falling back in as long again, and the drop (solid) falling from 100 m. For vertical motion the east–west velocity is always −2Ω cos λ times the height above the starting point, so the east–west drift is −2Ω cos λ times the area between the path and the starting height. The throw's area (lower shading) is (4/3)·h·t = 602 m s, above its start, so it drifts west; the drop's (upper shading) is h·t/3 = 151 m s, below its start, so it drifts east. The ratio of the two areas is four.

The ball thrown up that lands to the west

Drop a ball from a tower a hundred metres high and, because the Earth turns, it lands a centimetre and a half east of the point directly below. Throw a ball straight up so that it just reaches the same height, and it lands six centimetres west of where it was thrown — four times as far, the other way. The asymmetry has a one-line explanation that turns a problem in rotating frames into a problem about areas: for anything moving straight up or down, the sideways drift is proportional to the area under its graph of height against time. A dropped ball spends its flight below where it started, a thrown one above, and the thrown one's area is four times as large.

mechanics · Projectile

Named alongside it

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

TrajectoryEnvelopeOptimisationRangeReachable setParabolaReference frameStationary pointTangencyAngular momentumCausticConic section

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