Vector motion — Collision with $\tan\theta=3$

Question

Same projection setup as the exam (A from origin at angle , B horizontal from height with speed ). Given and the particles collide:

(i) By first showing , verify that .

(ii) Hence show that the collision time is .

(iii) When they collide their velocities are perpendicular. Show that (taking the earlier collision).

(iv) Hence find the height of the vertex of A's trajectory prior to collision, in terms of .

This practice question was inspired by Question 13(b) in the 2023 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: differentiate position to get velocity and acceleration, then translate the angle condition into a dot product.(i) . Triangle --: , . Thus .
Simplify the previous line carefully, keeping exact values where possible.(ii) .
Simplify the previous line carefully, keeping exact values where possible.(iii) Dot product zero: .
Simplify the previous line carefully, keeping exact values where possible.Discriminant : . Earlier collision uses the minus sign: .
Finish the calculation, then check that the result meets the question’s conditions.Then .
State the final answer clearly in the form the question requested.(iv) , so max height (rationalise).

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Same (i)–(iv) structure with ; perpendicular condition gives a quadratic in .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.