Mathematical Methods — Question 18

QCAA 2023, Paper 1 · 4 marks

Q18 · 2023 · Technology-freeComplex unfamiliar4 marks

QUESTION 18 (4 marks)

A person enters the lowest carriage of a miniature Ferris wheel with a six-metre diameter. The bottom carriage is one metre off the ground. When top speed is reached, it takes three seconds for a carriage to travel from the lowest to the highest point of the ride. It is claimed that:

**The vertical motion of the Ferris wheel produces a maximum vertical acceleration on each rider that is more than half the acceleration of free fall.**

Free fall occurs when gravity is the only force acting, resulting in an acceleration of

9.8\text{ m s}^{-2}

. Evaluate the reasonableness of the claim.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

4 marks

ANSWER

The claim is not reasonable; the maximum vertical acceleration is

\dfrac{\pi^2}{3}\approx3.29\text{ m s}^{-2}

, which is less than

4.9\text{ m s}^{-2}

Worked solution

Because of the repetitive nature of the motion, use a cosine graph (cosine since when

t=0

, the graph is a minimum) to model the motion. Let

h(t)=a\cos(b(t+c))+d.

The amplitude is

3

, the period is

6

seconds, so

b=\frac{2\pi}{6}=\frac\pi3

, there is no phase shift, and the axis is

4

m above ground. Hence

h(t)=-3\cos\left(\frac\pi3t\right)+4.

Then

v(t)=h'(t)=\pi\sin\left(\frac\pi3t\right),

and

a(t)=h''(t)=\frac{\pi^2}{3}\cos\left(\frac\pi3t\right).

The amplitude, and hence maximum acceleration, is

\frac{\pi^2}{3}\approx\frac{3^2}{3}=3.

Therefore, it is not reasonable to expect the acceleration of the Ferris wheel alone to be more than half of

9.8

(

4.9

correctly recognises a periodic model is to be used for the vertical position of the carriage

[1 mark]

correctly determines the model for the vertical position of a carriage on the Ferris wheel

[1 mark]

determines an approximation of the maximum acceleration produced based on the second derivative equation obtained

[1 mark]

provides appropriate statement of reasonableness

[1 mark]

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Mathematical Methods — Question 18

QUESTION 18 (4 marks)

QCAA guide · typeset solution

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