Ejercicio Exercise 1 · Física II URV 2024

Exercise 1 (10 points). Bungee jumper oscillation

An individual of mass $m = 60\text{ kg}$ leaps from a bridge situated at a height $h = 100\text{ m}$ above the ground, secured to an elastic rope of natural length $L_0 = 20\text{ m}$ and spring constant $k$. The jumper experiences free fall until the rope reaches its unstretched length. Beyond this point, the cord stretches and applies an upward force that decelerates the jumper until he momentarily halts and rebounds into oscillatory motion. Assume air resistance and the mass of the rope are negligible ($g = 9.8\text{ m/s}^2$).

a.

Calculate the spring constant $k$ required so that the jumper stops ($v = 0$) just before reaching the ground. [2 points]

b.

Determine the maximum elongation of this same rope if a second jumper of mass $m_2 = 90\text{ kg}$ performs the jump. [2 points]

c.

Compute the speed of this second jumper when reaching the ground level. [2 points]

d.

Establish the differential equation and solve for the equation of motion $x(t)$ of this second jumper once the rope begins to stretch, setting $x(0) = 0$ at the moment the rope starts extending. [4 points]