Q158 · Practice questionTechnology-freeComplex familiar6 marks
QUESTION 158 (6 marks)
A trolley moves on a straight track for seconds. Its velocity, in metres per second, is
The velocity-time graph is shown. The trolley starts at position .
a)[1 mark]
Determine the times at which the trolley reverses direction.
b)[3 marks]
Determine its displacement and total distance travelled over the four seconds.
c)[2 marks]
Determine whether the trolley ever moves to the negative side of its starting point. Justify your answer algebraically.
WORKED SOLUTION
6 marksPractice marking scheme
ANSWER
(a) Reversals at s. (b) Displacement m; distance m. (c) throughout; it returns to the start at s.
Worked solution
(a) changes sign at both and , so these are the reversal times.
(b) An antiderivative satisfying the initial position is
Thus , and . The displacement is m. Splitting at the reversals, the total distance is
(c) Factor the position: for . Therefore the trolley never lies on the negative side of its starting point, although its velocity is negative for . It returns to the start at .
Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Identify both sign-changing zeros at 1 and 3 s.
Part b: Integrate with the correct initial condition.
Part b: Obtain displacement 4/3 m.
Part b: Split at reversals to obtain distance 4 m.
Part c: Factor x(t)=t(t-3) squared /3.
Part c: Use non-negativity on the full domain to justify the conclusion.
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.