2021 HSC Mathematics Extension 1
Question 13(b):
Vector motion
When an object is projected from a point metres above the origin with initial speed at an angle of to the horizontal, its displacement vector seconds after projection is
(do NOT prove this).
A person, standing in an empty room which is high, throws a ball at the far wall of the room. The ball leaves their hand above the floor and from the far wall. The initial velocity of the ball is at an angle of to the horizontal.
Show that the ball will NOT hit the ceiling of the room but that it will hit the far wall without hitting the floor.
How to recognise this question
Question type: vector motion
- The command and notation point to vector motion.
- The requested response is a show that.
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
- Here , , . Vertical velocity: .
- Maximum height when : .
- .
- Since , the ball does not hit the ceiling.
- Time to floor (): .
- Positive root .
- Unrestricted range: .
- Alternatively at : , and .
- So the ball hits the far wall while still above the floor.
- NESA: 4 marks full; 3 for max height and time of flight (or equivalent); 2 for max height; 1 for vertical velocity expression.
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
With , , (same displacement model with -term ), find the maximum height.
Question 2
Using , , , find when and conclude whether the ball is still airborne.
Question 3
With , , room height , find the largest so that max height equals .