A body of density ρ' is dropped from rest at a height h into a lake of density ρ, where ρ > ρ'. Neglecting all dissipative forces, calculate the maximum depth to which the body sinks before returning to float on the surface.

Engineering

Physics

Conservation of Energy

Center of Mass

Buoyancy

Question

A body of density ρ' is dropped from rest at a height h into a lake of density ρ, where ρ > ρ'. Neglecting all dissipative forces, calculate the maximum depth to which the body sinks before returning to float on the surface.

$\frac{h ρ^{'}}{ρ}$

$\frac{h ρ^{'}}{ρ - ρ^{'}}$

$\frac{h ρ}{ρ - ρ^{'}}$

$\frac{h}{ρ - ρ^{'}}$

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A body of density ρ' falls from height h into a lake of density ρ (ρ > ρ'). It gains kinetic energy from the fall: $m g h$ . Underwater, buoyant force $ρ V g$ exceeds weight $ρ^{'} V g$ , causing deceleration. The net retarding force is $V g (ρ - ρ^{'})$ . Equating work done by this force to initial kinetic energy gives maximum depth d:

$V g (ρ - ρ^{'}) d = ρ^{'} V g h$

Solving for d: $d = \frac{ρ^{'} h}{ρ - ρ^{'}}$

Final answer: $\frac{h ρ^{'}}{ρ - ρ^{'}}$