Q157 · Practice questionTechnology-freeComplex familiar6 marks
QUESTION 157 (6 marks)
A particle moves for seconds with position vector, in metres,
Its path is shown. The point is its position at .
a)[1 mark]
Determine the Cartesian equation of the path.
b)[3 marks]
Determine the minimum and maximum speeds, and all times when each occurs.
c)[2 marks]
Determine all times when the position and velocity vectors are perpendicular.
WORKED SOLUTION
6 marksPractice marking scheme
ANSWER
(a) . (b) Minimum speed m/s at ; maximum m/s at . (c) .
Worked solution
(a) and , so .
(b) , giving
The minimum speed is 2 when , at . The maximum speed is 3 when , at . The endpoint is excluded.
(c) . It is zero exactly at the four times above. For an ellipse, the velocity is not perpendicular to the position vector at every point, even though it is tangent to the path.
Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Obtain the correct ellipse equation.
Part b: Differentiate and obtain speed squared 4+5 sin squared t.
Part b: Give minimum speed and both times.
Part b: Give maximum speed and both times.
Part c: Use the dot product to obtain -5 sin t cos t=0.
Part c: Give all four times in the half-open interval.
Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
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