Vector motion — Discriminant condition derivation

Question

Starting from for , show that the quadratic factor is positive for all precisely when .

This practice question was inspired by Question 14(d) in the 2024 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.Discriminant .
Finish the calculation, then check that the result meets the question’s conditions.Opening upwards with negative discriminant implies the quadratic is always positive.
State the final answer clearly in the form the question requested.Answer: (i.e. for ).

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

Verification: Leads to the critical angle .

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