What a mass ratio buys

At its defaults it draws what a mass ratio buys. Change of speed against the ratio of fuelled mass to dry mass, for exhaust speeds of 3.05 km/s, 4.44 km/s, 30 km/s. Each curve is the exhaust speed times the logarithm of the mass ratio, which is a shape with two unforgiving properties. A mass ratio of e buys exactly one exhaust speed and no more, whatever the engine; and doubling the achieved speed requires *squaring* the mass ratio, not doubling it. Reaching 9.4 km/s needs a mass ratio of 21.8 at 3.05 km/s of exhaust, 8.3 at 4.44 km/s of exhaust, 1.4 at 30 km/s of exhaust. The first of those is why a chemical rocket to orbit is nine parts propellant and one part everything else, and the last is why an ion engine, whose thrust would not lift its own weight, is nevertheless the only way of reaching the outer planets with a useful payload.

rocket-ledger 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.

Change of speed against the ratio of fuelled mass to dry mass, for exhaust speeds of 3.05 km/s, 4.44 km/s, 30 km/s. Each curve is the exhaust speed times the logarithm of the mass ratio, which is a shape with two unforgiving properties. A mass ratio of e buys exactly one exhaust speed and no more, whatever the engine; and doubling the achieved speed requires *squaring* the mass ratio, not doubling it. Reaching 9.4 km/s needs a mass ratio of 21.8 at 3.05 km/s of exhaust, 8.3 at 4.44 km/s of exhaust, 1.4 at 30 km/s of exhaust. The first of those is why a chemical rocket to orbit is nine parts propellant and one part everything else, and the last is why an ion engine, whose thrust would not lift its own weight, is nevertheless the only way of reaching the outer planets with a useful payload.

The mass of fuel a rocket must start with, divided by the mass it ends with, against the final speed as a fraction of light's — the vertical axis being the number of decades in that ratio, because the numbers do not fit on any other scale. Each solid curve is one exhaust speed, and each dashed one beside it is what Tsiolkovsky's Newtonian formula would have said. Below about a tenth of light speed the two are indistinguishable; above it they part, and the relativistic curve turns upward without limit as the final speed approaches light's, because what adds linearly is the rapidity rather than the velocity. The photon rocket — exhaust at exactly the speed of light, which is the best any engine can do — needs a mass ratio of 1.7321 to reach half light speed, which is √3 exactly, and 4.4 to reach nine tenths. Those are modest numbers and they are the whole of the good news. A chemical exhaust at 4 km/s needs 10^2968 even to reach a tenth of light speed, which is not a difficult engineering problem but an arithmetic impossibility — there are about 10⁵⁰ atoms in the Earth. What the chart cannot show is the other half of the trip: stopping at the far end squares the ratio, and coming home squares it again.

Change of speed against the ratio of fuelled mass to dry mass, for exhaust speeds of 3.05 km/s, 4.44 km/s, 30 km/s. Each curve is the exhaust speed times the logarithm of the mass ratio, which is a shape with two unforgiving properties. A mass ratio of e buys exactly one exhaust speed and no more, whatever the engine; and doubling the achieved speed requires *squaring* the mass ratio, not doubling it. Reaching 9.4 km/s needs a mass ratio of 21.8 at 3.05 km/s of exhaust, 8.3 at 4.44 km/s of exhaust, 1.4 at 30 km/s of exhaust. The first of those is why a chemical rocket to orbit is nine parts propellant and one part everything else, and the last is why an ion engine, whose thrust would not lift its own weight, is nevertheless the only way of reaching the outer planets with a useful payload.

Momentum against the fraction of the vehicle's initial mass that has been burnt, in units of that mass times a kilometre per second, for an exhaust speed of 3.05 km/s. The rising curve is the momentum of what is left of the vehicle; the falling one is the momentum of everything that has left it, each element carrying the speed it had when it was released, less the exhaust speed. Their sum is the flat line at zero, and it stays there to 5.8e-8 across the whole burn. That is the entire physics of a rocket. Nothing is pushed against, no medium appears in the arithmetic, and the vehicle accelerates for exactly the reason a person on a frictionless floor moves backwards on throwing a brick. The vehicle's own curve is not monotone in an obvious way either: it rises because the speed is growing faster than the mass is falling, and it would turn over if the burn continued past a mass ratio of e.

The mass of fuel a rocket must start with, divided by the mass it ends with, against the final speed as a fraction of light's — the vertical axis being the number of decades in that ratio, because the numbers do not fit on any other scale. Each solid curve is one exhaust speed, and each dashed one beside it is what Tsiolkovsky's Newtonian formula would have said. Below about a tenth of light speed the two are indistinguishable; above it they part, and the relativistic curve turns upward without limit as the final speed approaches light's, because what adds linearly is the rapidity rather than the velocity. The photon rocket — exhaust at exactly the speed of light, which is the best any engine can do — needs a mass ratio of 1.7321 to reach half light speed, which is √3 exactly, and 4.4 to reach nine tenths. Those are modest numbers and they are the whole of the good news. A chemical exhaust at 14990 km/s needs 10^1 even to reach a tenth of light speed, which is not a difficult engineering problem but an arithmetic impossibility — there are about 10⁵⁰ atoms in the Earth. What the chart cannot show is the other half of the trip: stopping at the far end squares the ratio, and coming home squares it again.

The thrust of one F-1 engine against ambient pressure, in atmospheres. It is 6.78 meganewtons at sea level and 7.77 in vacuum — 14.6 per cent more with the air taken away. The line falls at exactly 9.787 newtons per pascal, which is the nozzle's exit area, because the term is (p_e − p_a)·A_e: the ambient pressure pushes on the exit plane from outside and there is nothing to push back on it. The account in which the exhaust shoves against the atmosphere makes the opposite prediction — thrust falling with the pressure and vanishing in vacuum — and it is not a small disagreement about a coefficient. It has the sign wrong. A rocket works better in vacuum than in air, and every engine ever fired has said so.

physicscheck asserts something about rocket-ledger 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.