2025 HSC Mathematics Extension 1
Question 13(c):
Vector motion
At time , a particle has position vector , velocity and acceleration . Find the positive time when the angle between and is .
How to recognise this question
Question type: vector motion
- The command and notation point to vector motion.
- The requested response is a worked solution.
How to handle it: differentiate position to get velocity and acceleration, then translate the angle condition into a dot product.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Differentiate : , .
- .
- Also .
- Equate, square and solve to get ; since time is positive, .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 4 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
For , find velocity and acceleration at .
Question 2
For and , find positive t when they are perpendicular.
Question 3
A particle has and . Find .