A man of height `h_(0)=2 m` is bungee jumping from a platform situated at a height h = 25 m above a lake One end of an elastic rope is attached to his foot and the other end is fixed to the platform He starts falling from rest in vertical position. The length and elastic properties of the rope are chosen so that his speed will have been reduced to zero at instant when his head reaches the surface of water. Ultimetely the jumper is hanging from the rope with his head 8m above the water. Find the maxima acceleration acheived during the jump in `m//s^(2))`

Answer

`40`

Explanation

Let us denote the elastic constant (spring constant) of the rope by k and its unstretched length by `l_(0)` The maximum length of the rope is `l_(1)=h-h_(0)=23` m whilst in equilibrium it is `l_(2)=(23-8)m=15m` initially and at the jumper's lowest position, the kinetic energy is zero if we ignore the mass of the rope and assume that the jumper's centre of mass is half - way up his body, we can use conservation of energy to write `mgh=1/2k(l_(1)-l_(0))^(2)`
In addition in equilibrium `mg=k(l_(2)-l_(0))`
Dividing the two equations by each other we obtain a quadratic equation for l_(0)
`l_(0)^(2)+2(h-l_(1))l_(0)+(l_(1)^(2)-2hl_(2))=l_(0)^(2)+4l_(0)-221=0`
which gives `l_(0)=13m`
When the falling jumper attains his highest speed, his acceleration must be zero and so this much occurat the same level as the final equilibrium position `(l=l_(2))`
Again applying the law of conservation of energy
`1/2mv^(2)+1/2k(l_(2)-l_(0))^(2)=mg(l_(2)+h_(0))`
where the ratio m//k is the same as that obtained from the equilibrium condition, namely
`m/k = (l_(2)-l_(0))/g`
Substituting this into the energy equation, shows that the maximum speed of the jumper is `v=18 ms^(-1)approx 65km h^(-1)` It is easy to see that his maximum acceleration occurs at the lowest point of the jump Since the largest extension of the rope (10m) is five times that at the equilibrium position (2m), the greatest tension in the rope is 5 mg So the highest net force exerted on the jumper is 4 mg and his maximum acceleration is 4g.