# Question #14ee4

Feb 19, 2017

Gravitational Potential Energy of air bubble increases when it rises up.
Buoyancy Potential Energy decreases as the bubble rises.

#### Explanation:

As a bubble rises through a fluid three forces act upon it as shown in the figure below For an air bubble the interface is water only, as such drag is zero. As the bubble is rising there is net upwards force acting on the bubble.

${\vec{F}}_{\text{net"=vecF_"Buoyancy"-vecF_"Gravity}}$

It may be seen that Gravitational potential energy increases as the bubble of mass $m$ rises, or height $h$ increases as

$\Delta P {E}_{\text{Gravitation}} = m g \Delta h$.

Also due net upwards force there is upward motion of velocity $v$ and bubble has kinetic energy

$K E = \frac{1}{2} m {v}^{2}$

As the bubble rises, pressure on the bubble decreases, which leads to expansion of bubble and thereby increase of force due to buoyancy. As such kinetic energy increases.

Another energy which is associated with the bubble is its Buoyancy potential energy which is directly proportional to the depth $d$ of the bubble.

$\Delta P {E}_{\text{Buoyancy"=rho_wV_"Bubble}} g \Delta d$
where ${\rho}_{w}$ is density of water, ${V}_{\text{Bubble}}$ is volume of bubble, $g$ acceleration due to gravity.

As the bubble rises its depth decreases. Hence Buoyancy potential energy decreases.

As a result of the Law of Conservation of energy:
The increase in Kinetic and Gravitational Potential energies is due to decrease in Buoyancy potential energy of bubble.