Mid-air Collision
Two objects launched toward each other: do they meet in the air, and if not, how close do they get? Or let the calculator find the angle or speed that makes object A hit object B — then press play and watch.
Why Gravity Drops Out
Gravity pulls both objects down by exactly the same amount at every moment. So seen from object A, object B doesn't fall at all: it moves in a straight line at constant speed. Whether they collide is a question of straight-line geometry, not of parabolas.
That explains the classic monkey and hunter puzzle: a hunter aims straight at a monkey hanging in a tree, and the monkey lets go the moment the gun fires. The dart and the monkey fall together, so the dart hits — at any speed, as long as it gets there before the monkey reaches the ground.
Trajectories
to scaleCollision Formulas
both in flight| Quantity | Formula |
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Worked Techniques
Subtract the two position formulas: the ½gt² terms cancel. What's left is a straight line, which makes every collision question much easier.
The distance along a straight line is smallest where the separation is perpendicular to the relative velocity. Check that both objects are still in the air at t*.
Aim straight at a target that drops the moment you launch: you hit it after time t, at a point ½gt² below where it hung — provided that point is still above the ground.
Sources & Verification
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014 — Ch. 4 (Motion in Two and Three Dimensions).