Vector motion — Collision with $\tan\theta=2$

Question

Particle A is projected from the origin with speed at angle to the horizontal. Particle B is projected horizontally with speed from height above the origin at the same instant. Position vectors: , . Given and the particles collide:

(i) By first showing , verify that .

(ii) Hence show that the particles collide at time .

(iii) When they collide their velocity vectors are perpendicular. Show that .

(iv) Hence find the maximum height of A's parabolic 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.

Step by step

Set up the solution: differentiate position to get velocity and acceleration, then translate the angle condition into a dot product.(i) Equate -coordinates: . With : , so .
Simplify the previous line carefully, keeping exact values where possible.(ii) Equate -coordinates: . With : .
Simplify the previous line carefully, keeping exact values where possible.(iii) at : .
Finish the calculation, then check that the result meets the question’s conditions.With : .
State the final answer clearly in the form the question requested.(iv) Vertex of A when : . Height .

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

Verification: Full four-part dependent chain as in the 2023 paper.

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

← Back to 2023 HSC Extension 1 paper Question 13(b)