Q167 · Practice questionTechnology-freeComplex unfamiliar7 marks
QUESTION 167 (7 marks)
A projectile is launched from the origin at with velocity metres per second and constant acceleration metres per second squared.
A target moves with position vector
The projectile model applies until it first returns to ground level . The diagram shows its path and the target's horizontal path.
a)[2 marks]
Determine the projectile's position vector and its flight-time interval.
b)[3 marks]
Determine whether the projectile hits the moving target. If it does, give the time and position.
c)[2 marks]
Determine the maximum height and state whether the projectile is ascending or descending at interception.
WORKED SOLUTION
7 marksPractice marking scheme
ANSWER
(a) , . (b) Interception at s, position m. (c) Maximum height 18 m at s; interception occurs while descending.
Worked solution
(a) Integrating acceleration and using the initial velocity gives . Integrating again and using the origin gives . Its positive ground-return time solves , so the physical interval is .
(b) Matching horizontal coordinates requires , giving . At that time , and both horizontal coordinates are 40. Thus they meet at at 5 s, within the flight interval. Matching only the two paths would not establish simultaneous interception.
(c) Vertical velocity is zero at , giving . The height there is m. At interception the vertical velocity is m/s, so the projectile is descending.
Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Integrate with the initial conditions to obtain the position vector.
Part a: Obtain the flight interval 0 to 6 s.
Part b: Match same-time horizontal positions to obtain t=5.
Part b: Verify that vertical positions also agree.
Part b: State interception at (40,10) at 5 s within the flight interval.
Part c: Obtain maximum height 18 m at t=3.
Part c: Use vertical velocity -8 m/s at t=5 to justify descent.
Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
View the QCAA syllabusHow many marks did you earn?
Compare your working with the guide above.