Illustrated Physics
physics, drawn
Concept

Parabola — where it appears

The curve of points equally distant from a fixed point, the focus, and a fixed line, the directrix; algebraically, the graph of a quadratic. It is the path of a projectile without air resistance and the shape that brings parallel rays to a single focus.

Named by 3 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 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.

mechanics · Projectile

Named alongside it

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

EnvelopeProjectileRangeReachable setStationary pointTrajectoryApproximationCausticConic sectionDiscriminantEllipseFocus

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