After earning an 'A' in PHY 183 you land a job with ACME Bungee Jump company. They need to know what spring stiffness k_S to make the cords so that a jumper of mass 200kg will only fall 30m. The standard length of an un-stretched cord is just 10m.

Jumper of mass 200kg

Jumper will fall 30m

Standard length of un-stretched cord is 10m

Initial State: Jumper motionless at top

Final State: Jumper motionless 30m below

k_s

Jumper is motionless before they start

No energy lost to the surroundings

System: Jumper + Cord + Earth

Surroundings: Nothing

course_planning

$$\Delta E_{system} = W_{surroundings}$$

$$\Delta K_{jumper} + \Delta U_{grav} + \Delta U_{spring} = W_{surroundings} = 0$$

$$\Delta K_{jumper}$$ is 0 because jumper is motionless before + after jumpW_{surroundings} $$no work$$ \Delta U_{grav} = -\Delta U_{spring}mg(y_{f}-y_{i}) = \dfrac{1}{2}ks({s_{f}}^2 - {s_{i}}^2)$$