Mathematical Methods — Question 18

QCAA 2024, Paper 2 · 5 marks

Q18 · 2024 · Technology-activeComplex unfamiliar5 marks

QUESTION 18 (5 marks)

An object experiencing straight-line motion along a path has an acceleration (m s

^{-2}

) defined by the function

a(t)=3\sin(2t)

where

t

is time (s) since the object begins moving,

t\ge0

. When

t=0

, both displacement and velocity are zero. On the path is a motion sensor that is able to detect motion up to 2 metres away. The object passes directly by the motion sensor when

t=3

. Determine the average velocity of the object while it moves through the range of the sensor.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

5 marks

ANSWER

Approximately

1.378

m/s.

Worked solution

Integrating acceleration and using

v(0)=0

gives

v(t)=-\frac32\cos(2t)+\frac32.

Integrating again and using

d(0)=0

gives

d(t)=-\frac34\sin(2t)+\frac32t.

t=3

d(3)=4.70956\text{ m}.

Thus the sensor range is from

2.70956

m to

6.70956

m. Solving

d(t)=2.70956

and

d(t)=6.70956

gives entry and exit times

t=1.68914\text{ s},\qquad t=4.59214\text{ s}.

The object travels 4 m through the range in

4.59214-1.68914=2.90296\text{ s},

so its average velocity is

\frac4{2.90296}=1.378\text{ m/s}.

correctly determines the velocity function

[1 mark]

correctly determines the displacement function

[1 mark]

determines the sensor position and/or detection-range positions

[1 mark]

determines the times at which the object enters and leaves the detection range

[1 mark]

determines the average velocity while within the detection range

[1 mark]

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Mathematical Methods — Question 18

QUESTION 18 (5 marks)

QCAA guide · typeset solution

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