Will These Two Planes Get Too Close?
Two aircraft fly straight paths through the same airspace. From their positions and speeds, work out how close they'll really get — and design the smallest course change that keeps them apart.
Where this idea comes from
Start here — these are the sources that inspired this exploration.
Introduction
Every airliner carries a system called TCAS that watches nearby aircraft and predicts how close each one will come. At its heart is a question you can answer with school maths: two objects move in straight lines at constant speed — what is the minimum distance between them, and when does it happen? Set each plane up with a starting position and a velocity, write where each one is at time t, and study the distance between them. Then play air-traffic controller: when the answer is too close for comfort, what is the gentlest instruction that fixes it?
Guiding Questions
- Start flat: two paper planes on a grid, each with a position and a velocity. Where is each one after t seconds, and what's the distance between them as t grows?
- The distance shrinks, bottoms out, and grows again. Find the minimum exactly — and check it against a table of values.
- Add altitude: real aircraft separation is 3D, and the rules treat vertical and horizontal distance differently. How does your answer change?
- Suppose your minimum is too close for comfort. Choose the smallest change to one flight that restores safe separation, and defend why yours is the gentlest option.
- Alter the encounter — your choice of what — and investigate how sensitive the closest approach is to it. What does that tell you about why TCAS keeps recalculating?