TestMum
← 2023 HSC Extension 1 questions

2023 HSC Mathematics Extension 1

Question 13(b):
Vector motion

8 marksvector motiondependency-safe group

Particle A is projected from the origin with speed at angle to the horizontal. At the same time particle B is projected horizontally with speed from height above the origin.

Position vectors (Do NOT prove): , .

Given and the particles collide:

(i) By first showing , verify that .

(ii) Show that the particles collide at time .

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

(iv) Prior to collision, the trajectory of A is a parabola. Find the height of the vertex of that parabola above the horizontal plane, in terms of .

Particle A is projected from O at speed v and angle theta, while particle B is projected horizontally at speed u from height H.
Particle A is projected from O at speed v and angle theta, while particle B is projected horizontally at speed u from height H.

How to recognise this question

Question type: vector motion

  • The command and notation point to vector motion.
  • The word “hence” means the later part is intended to reuse the earlier result.

How to handle it: differentiate position to get velocity and acceleration, then translate the angle condition into a dot product.

Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.

Step-by-step answer

  1. (i) Collision equates -coordinates: (for ).
  2. With in first quadrant: opposite , adjacent , hypotenuse , so , .
  3. Thus .
  4. NESA (i): 2 marks; 1 for equating -coordinates.
  5. (ii) Equate -coordinates at collision time : .
  6. Substitute and : .
  7. NESA (ii): 1 mark.
  8. (iii) Velocities: , .
  9. Perpendicular at : .
  10. Use , : .
  11. With : .
  12. NESA (iii): 3 marks; 2 for evaluating the dot product set to ; 1 for observing perpendicularity means dot product zero.
  13. (iv) For particle A, . Vertex when : .
  14. Height: .
  15. NESA (iv): 2 marks; 1 for setting vertical velocity of A to zero.

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 8 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

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 .

See the step-by-step answer guide →

Question 2

Same projection setup as the exam (A from origin at angle , B horizontal from height with speed ). Given and the particles collide:

(i) By first showing , verify that .

(ii) Hence show that the collision time is .

(iii) When they collide their velocities are perpendicular. Show that (taking the earlier collision).

(iv) Hence find the height of the vertex of A's trajectory prior to collision, in terms of .

See the step-by-step answer guide →

Question 3

Same projection setup. Given and the particles collide:

(i) Show and hence .

(ii) Hence show the collision time is .

(iii) Velocities are perpendicular at collision. Taking the earlier collision time, show .

(iv) Hence express the vertex height of A's trajectory in terms of .

See the step-by-step answer guide →