Vector motion — Collision with $\tan\theta=4$

Question

Same projection setup. Given and the particles collide:

(i) Show and hence .

(ii) Hence show the collision time is .

(iii) Velocities are perpendicular at collision. Taking the earlier collision time, show .

(iv) Hence express the vertex height of A's trajectory 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) ; , , so .
Simplify the previous line carefully, keeping exact values where possible.(ii) .
Simplify the previous line carefully, keeping exact values where possible.(iii) .
Finish the calculation, then check that the result meets the question’s conditions.Earlier root: , so .
State the final answer clearly in the form the question requested.(iv) Max height .

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

Verification: Full four-part chain; still yields real perpendicular collisions ().

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