SSCE HSC Mathematics Extension 2 Resisted vertical motion: 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. 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

$$ \ddot{x} = -g - rv $$

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

$$ x = \frac{g}{r} \ln \left( \frac{g}{g - rv} ight) - \frac{v}{r} $$

(2) Given that g = 9.8 m s^-2 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 x is given by

$$ x = e^{t/10} (29 \sin t - 10 \cos t) + 92 $$

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.

(b) Let z = \cos 	heta + i \sin 	heta .

(i) Show that $ z^2 + z^{-2} = 2 \cos n 	heta $, where n is a positive integer.

(ii) Let m be a positive integer. Show that

$$ (2 \cos 	heta)^2 m = 2 \cos 2m \left( \sum_{k=1}^{m} \frac{(2m - 2k)(2m - 4k)}{(2k)(2m - 2k)} ight) + (2m) $$

(iii) Hence, or otherwise, prove that

$$ \int_0^{\pi} \cos^{2m} 	heta d	heta = \frac{\pi}{2^{2m + 1}} $$

where m is a positive integer.

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 s^-2 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.

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

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