Vector motion — A perpendicular-velocity time

Question

For and , find positive t when they are perpendicular.

This practice question was inspired by Question 13(c) in the 2025 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.Perpendicular vectors have dot product zero.
Simplify the previous line carefully, keeping exact values where possible..
Finish the calculation, then check that the result meets the question’s conditions., so .
State the final answer clearly in the form the question requested.Final answer:

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

Verification: Substitution makes the dot product zero.

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