Generator

Trajectories at one speed and several angles

One function in the mechanics library, called 36 times across 6 essays. Below: what it draws at its defaults, what it draws at every branch an essay asks for, whether the site's own gate puts a claim to it, and everywhere it is called.

At its defaults it draws 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.

projectile-angles is one function in lib/figures/mechanics.js — motion, force, energy and rotation. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.

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.

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 boundary of everywhere a given speed can reach

The options are the ones Everywhere a throw can reach passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

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.

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.

Two ways to hit anything inside it, and no way to hit anything outside

The options are the ones Everywhere a throw can reach passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Two ways to hit anything inside it, and no way to hit anything outside. The reachable set of a projectile launched at one speed in any direction, shaded, with its boundary. A point inside it is reached by exactly two launches — a low one and a high one, here 65.6° and 45.5° for the marked target — because the range equation is a quadratic in the tangent of the launch angle and a quadratic has two roots. On the boundary the two roots coincide, which is what a tangency is. Outside it the discriminant is negative and there is no launch angle at all, real or otherwise: the second marked point is a fifth of a v²/g above the boundary and no aiming reaches it. Artillery tables are two-valued for this reason, and the high trajectory is chosen when something has to be cleared and the low one when the flight has to be short.

The reachable set of a projectile launched at one speed in any direction, shaded, with its boundary. A point inside it is reached by exactly two launches — a low one and a high one, here 65.6° and 45.5° for the marked target — because the range equation is a quadratic in the tangent of the launch angle and a quadratic has two roots. On the boundary the two roots coincide, which is what a tangency is. Outside it the discriminant is negative and there is no launch angle at all, real or otherwise: the second marked point is a fifth of a v²/g above the boundary and no aiming reaches it. Artillery tables are two-valued for this reason, and the high trajectory is chosen when something has to be cleared and the low one when the flight has to be short.

Trajectories at one speed and several angles

The options are the ones Everywhere a throw can reach passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

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.

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.

Range against launch angle, at five drags

The options are the ones Everywhere a throw can reach passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Range against launch angle, at five drags. Range against launch angle for ballistic parameters of 0, 0.3, 1.2, 4, 12, each curve integrated point by point. The top curve is the vacuum case, symmetric about 45° because sin 2θ is, and every other curve is asymmetric: the peak moves left as the drag rises — to 45° at 0, 42.5° at 0.3, 40° at 1.2, 35° at 4, 32.5° at 12 — and the fall-off is much steeper on the high side than on the low. That asymmetry is the practical content. Throwing ten degrees under the optimum costs almost nothing in air; throwing ten degrees over it costs a great deal, because the extra height is bought with horizontal speed that drag then removes. Every curve is also lower than the one above it, which is the loss, but the loss is not what moves the peak.

Range against launch angle for ballistic parameters of 0, 0.3, 1.2, 4, 12, each curve integrated point by point. The top curve is the vacuum case, symmetric about 45° because sin 2θ is, and every other curve is asymmetric: the peak moves left as the drag rises — to 45° at 0, 42.5° at 0.3, 40° at 1.2, 35° at 4, 32.5° at 12 — and the fall-off is much steeper on the high side than on the low. That asymmetry is the practical content. Throwing ten degrees under the optimum costs almost nothing in air; throwing ten degrees over it costs a great deal, because the extra height is bought with horizontal speed that drag then removes. Every curve is also lower than the one above it, which is the loss, but the loss is not what moves the peak.

The best launch angle, against how much drag there is

The options are the ones Everywhere a throw can reach passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

The best launch angle, against how much drag there is. The launch angle of greatest range against the ballistic parameter — the drag force at launch measured in weights — over nearly four decades, each point found by integrating the trajectory and searching. At the left the answer is 45.01°, which is the vacuum result recovered rather than assumed. It falls monotonically from there: 44.9° for shot put, 38.1° for golf ball, drag only, 28.7° for shuttlecock. The reason is asymmetry rather than loss. Drag takes most from the fastest part of the flight, so a launch spends its speed early; a lower angle keeps more of that early speed horizontal, where it buys range, and the height that a steeper launch buys is returned at a descent speed that drag has already capped. There is no formula on this chart. With quadratic drag the equations do not separate and there is no closed-form trajectory, so every point here is an integration.

The launch angle of greatest range against the ballistic parameter — the drag force at launch measured in weights — over nearly four decades, each point found by integrating the trajectory and searching. At the left the answer is 45.01°, which is the vacuum result recovered rather than assumed. It falls monotonically from there: 44.9° for shot put, 38.1° for golf ball, drag only, 28.7° for shuttlecock. The reason is asymmetry rather than loss. Drag takes most from the fastest part of the flight, so a launch spends its speed early; a lower angle keeps more of that early speed horizontal, where it buys range, and the height that a steeper launch buys is returned at a descent speed that drag has already capped. There is no formula on this chart. With quadratic drag the equations do not separate and there is no closed-form trajectory, so every point here is an integration.

What checks it

physicscheck asserts something about projectile-angles that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

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.

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.

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.

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.

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.

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.

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