Try this and see
Three experiments where the obvious prediction is wrong
Every figure below comes from running this simulation, not from theory. Load each pair, watch the readout, and check the prediction before you read the result.
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1Navigation gain N: 3 → 5, nothing else touched
- What most people predict
- A higher gain is a stronger guidance law, so the intercept should come noticeably sooner and cleaner.
- What the tool prints
- Both runs read INTERCEPT at 10.8 s. The clock cannot tell them apart. What changes is the acceleration trace, whose peak rises by about two thirds.
- Why
- N does not decide whether the line of sight stops drifting; it decides how hard you pull to stop it. The geometry already fixes where the meeting point is, so the flight time is settled before N is chosen. All the extra gain buys is a harder pull — and structural load is the one thing a real airframe cannot renegotiate.
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2Against the 9 g weaving target, g-limit: 8 → 9
- What most people predict
- More available g should help gradually — each extra g of capability shaving a bit off the miss distance.
- What the tool prints
- That is exactly what happens, and the verdict line hides it. Closest approach falls 0.85 → 0.71 → 0.56 → 0.41 km as you step 8 → 8.25 → 8.5 → 8.75 g: a straight line, 0.15 km per quarter-g. But the outcome says MISS every time, until 9 g, where it reads INTERCEPT at 11.1 s.
- Why
- Nothing sudden happens at 9 g. The run counts as an intercept when the range drops below 0.4 km, so the smooth curve simply crosses that line there. Read the closest-approach figure, not the verdict, and you can see the trend the pass/fail label throws away — which is the difference between knowing you failed and knowing by how much.
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3Interceptor speed: 1500 → 800 m/s, against a 1100 m/s target
- What most people predict
- An interceptor slower than its target cannot catch it. This one is only 73% of the target speed, so it should trail behind and miss.
- What the tool prints
- INTERCEPT at 16.8 s — against 10.2 s at full speed. It is late, not beaten. Drop to 750 m/s and it finally fails: MISS — closest approach 0.50 km.
- Why
- Proportional navigation does not chase; it flies to where the target will be. In a crossing engagement the interceptor cuts the corner, so it has to arrive at the meeting point at the same moment — not out-run anything. Being slower costs margin and time, and only below about 750 m/s does it cost the intercept.
References (3)
- The guidance law the tool implements, and where N = 3–5 comes from: P. Zarchan, Tactical and Strategic Missile Guidance, 6th edition, ch. 2 and 8. AIAA, 2012. ISBN 978-1-60086-894-8.
- Insight block 3 — the dragonflies: M. Mischiati, H.-T. Lin, P. Herold, E. Imler, R. Olberg and A. Leonardo, "Internal models direct dragonfly interception steering." Nature 517, 333–338, 2015.
- Proportional navigation, in the paper that introduced it to the open literature: L. C.-L. Yuan, "Homing and Navigational Courses of Automatic Target-Seeking Devices." Journal of Applied Physics 19(12), 1122–1128, 1948.
Problem solved in full
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Lateral acceleration for an interceptor flying 1500 m/s with navigation ratio N = 3 8 steps
An interceptor at (22, 2) km flying 1500 m/s, a target at (5, 19) km flying 900 m/s on a heading of -14°, and the interceptor launched straight down the line of sight. With navigation ratio N = 3, what lateral acceleration does proportional navigation demand in the first instant — and why does it demand anything at all, when the missile is already pointed exactly at the target?
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Subtract the two positions: 17 km to the left and 17 km up. The 45° symmetry is a gift, because R² comes out as exactly 578 and every step below divides by it.
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The target is not manoeuvring in this engagement, so its velocity vector never changes. Resolve it once and reuse it.
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The interceptor is launched along the line of sight, so its heading ψ0 is 135° as well. Subtract the two velocities and you have the only vector the guidance law ever looks at.
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Closing velocity is the part of the relative velocity lying along the line of sight, with the sign flipped so that closing counts positive. Only that part shortens the range.
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What is left of the relative velocity points across the line of sight, and that is what swings the bearing. Divide it by the range for the rate. It is not zero — so despite the perfect aim, this is not a collision course.
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Proportional navigation multiplies those two readings together, scales by N, and calls the product an acceleration. Nothing in the law knows where the target is going; it knows how fast the bearing is drifting and how fast the range is closing, and that is all.
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Now get the same rate a second way, with no dot products at all. Resolve the target's own velocity perpendicular to the line of sight.
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The interceptor contributes nothing across the line of sight, because it was launched along it. Every bit of the 463.5 m/s of cross-bearing drift belongs to the target, and dividing by the 24,042 m of range reproduces step 5 exactly.
Answer
131.4 m/s², or 13.4 g, demanded of a missile pointed precisely at its target. Aiming at the target is the one heading guaranteed to be wrong against anything moving sideways. A real collision course needs the interceptor to carry the same 463.5 m/s across the line of sight that the target carries, and at 1500 m/s that means aiming 18.0° off it — heading 117°, not 135°. Proportional navigation never solves that triangle. It measures the drift, multiplies by N and Vc, and turns: the first command bends the flight path at 131.4/1500 = 0.0876 rad/s, or 5.02°/s. At that rate the missing 18° of lead costs 3.6 s to buy, and the command shrinks as the drift it is cancelling shrinks, so the real swing takes longer still. Raise N and the same 18° is bought sooner at proportionally more g, which is the whole of the trade the navigation ratio controls.
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