Vector motion — Why differentiate D squared

Question

Explain why showing for proves that the distance is increasing, then state the critical angle condition in terms of .

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.Since for and , the signs of and agree.
Finish the calculation, then check that the result meets the question’s conditions.The condition is equivalent to .
State the final answer clearly in the form the question requested.Answer: ensures is increasing for all .

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

Verification: Standard projectile distance argument.

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