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Example: The Jumper
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.
Facts
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
Lacking
k_s
Approximations & Assumptions
Jumper is motionless before they start
No energy lost to the surroundings
Representations
Solution
$$\Delta E_{system} = W_{surroundings}$$
$$\Delta K_{jumper} + \Delta U_{grav} + \Delta U_{spring} = W_{surroundings} = 0$$
$$\Delta K_{jumper}$$ is 0 because jumper is motionless before + after jump$$ $$W_{surroundings}$$ no work $$\Delta U_{grav} = -\Delta U_{spring}$$ $$mg(y_{f}-y_{i}) = \dfrac{1}{2}ks({s_{f}}^2 - {s_{i}}^2)$$