Projectile — the series · Illustrated Physics

Trajectories at one speed and several angles. Projectile paths launched at the same speed and five different angles. The 45° launch travels furthest, and the 20° and 70° launches land in the same place.

The angle that throws furthest, and why nobody notices

Forty-five degrees is the answer, and the maximum is so flat that a throw ten degrees off loses almost nothing. Both halves of that are worth drawing.

part 1 · mechanics

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.

part 2 · mechanics

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.

part 3 · mechanics

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.

part 4 · mechanics

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.

part 5 · mechanics

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.

part 6 · mechanics

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.

part 7 · mechanics

Throws that grow into orbits. The longest throws possible from one point on the Earth, at launch speeds of 3, 5, 6.5, 7.5 km/s, ignoring air, each at its best angle, drawn to scale round the Earth. 3 km/s: launched at 42.8° above the horizontal, landing 1.0 thousand km away; 5 km/s: launched at 37.8° above the horizontal, landing 3.2 thousand km away; 6.5 km/s: launched at 29.7° above the horizontal, landing 6.8 thousand km away; 7.5 km/s: launched at 17.6° above the horizontal, landing 12.2 thousand km away. The faster the throw, the flatter the best launch and the further round the Earth it lands; at the circular-orbit speed, 7.91 km/s, the best launch is horizontal and the throw never comes down.

The parabola that is the top of an ellipse

Every textbook throw is a parabola, the best angle is 45°, and the range goes as the square of the speed. All three are statements about a flat Earth with gravity pointing the same way everywhere. On the real one a throw is a piece of an orbit — an ellipse with the Earth's centre at one focus — and the parabola is what the top of that ellipse looks like when the Earth is large. Throw faster and the best angle falls, the range runs ahead of the formula, and at eight kilometres a second the best throw is horizontal and never lands.

part 8 · mechanics

Three throws in a spinning room. Balls thrown at 5 m/s from the floor of a station 10 m in radius, spun at 0.99 rad/s (9.5 turns a minute) so that its floor presses with one g, as seen by someone standing in it: at 45° spinward, at 45° antispinward and straight up; with the Earth's parabolas for the two 45° throws dashed. Distances along the floor and heights in metres. On the Earth both 45° throws land 2.55 m away. In the station the spinward throw lands 1.52 m away and the antispinward one 3.06 m away, and the ball thrown straight up comes down 1.30 m ahead of the thrower, spinward. Nothing acts on any of the balls in flight. Each flies straight across the station while the floor turns beneath it, and what the thrower sees as gravity bending the path is the room rotating.

The throw a spinning station bends

A space station spun fast enough presses its floor up against its crew with the force of the Earth's gravity, and from the inside it can feel like home. Throw a ball, and the illusion fails. Nothing acts on the ball once it leaves the hand: it flies in a perfectly straight line across the station while the floor turns beneath it, and what the thrower sees as a parabola is a chord of a circle seen from a turning room. In a station ten metres across, a ball thrown up a metre comes down a metre ahead of the hand, a ball thrown with the spin falls short and one thrown against it carries long, and a dropped spanner lands a metre behind the point beneath it. How big a station must be to feel like the Earth is a question projectiles answer.

part 9 · mechanics