SSCE HSC Mathematics Extension 2 De Moivre’s theorem: A bungee jumper of height 2 m fa

Question

A bungee jumper of height 2 m falls from a bridge which is 125 m above the surface of the water, as shown in the diagram. The jumper's feet are tied to an elastic cord of length, L m. The displacement of the jumper's feet, measured downwards from the bridge, is x m.

The jumper's fall can be examined in two stages. In the first stage of the fall, where 0 ≤ x ≤ L, the jumper falls a distance of L m subject to air resistance, and the cord does not provide resistance to the motion. In the second stage of the fall, where x > L, the cord stretches and provides additional resistance to the downward motion.

(i) The equation of motion for the jumper in the first stage of the fall is
$$\frac{d^2x}{dt^2} = -g - rv$$
where g is the acceleration due to gravity, r is a positive constant, and v is the velocity of the jumper.

(1) Given that x = 0 and v = 0 initially, show that
$$x = \frac{g}{2r} \ln \left( \frac{g}{g - rv} \right) - \frac{v}{r}$$

(2) Given that g = 9.8 m s² and r = 0.2 s⁻¹, find the length, L, of the cord such that the jumper’s velocity is 30 m s⁻¹ when x = L. Give your answer to two significant figures.

(ii) In the second stage of the fall, where x > L, the displacement is given by
$$x = e^{\frac{t}{10}} (29 \sin t - 10 \cos t) + 92$$
where t is the time in seconds after the jumper's feet pass x = L.
Determine whether or not the jumper’s head stays out of the water.

A bungee jumper of height 2 m falls from a bridge which is 125 m above the surface of the water, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 7 - 2009 - Paper 1

Worked Solution & Example Answer:A bungee jumper of height 2 m falls from a bridge which is 125 m above the surface of the water, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 7 - 2009 - Paper 1

Given that g = 9.8 m s² and r = 0.2 s⁻¹, find the length, L, of the cord such that the jumper’s velocity is 30 m s⁻¹ when x = L. Give your answer to two significant figures.

Determine whether or not the jumper's head stays out of the water.

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