# How to calculate speed of buoyant object rising

I'm looking for a formula to calculate the speed of object in buoyancy. Is there any formula available for this purpose?

For example, if I release one liter water bottle from under 10 meter distance of water surface. In which speed the bottle can travel to reach water surface (like m/s)?

NOTE: Bottle filled with 1 liter air.

I'm sorry for my English. If there is any grammar mistakes on this question. Please edit my question

We can arrive at a simple formula for a maximum speed from some reasonable assumptions: An object with mass $m$ and volume $V$ , released from rest fully immersed in a fluid of constant density $\rho$, the initial net force is:

$$F = -mg+\rho Vg$$ where $F$ is assumed to point upward, and therefore the objects density ($\frac mV$) to be lower than that of the fluid, and $g$ is the gravitational acceleration.

As the object starts to rise, it experiences fluid resistance, reducing its acceleration. Eventually a balance is reached:

$$0 = F = -mg+\rho Vg - \frac 12 \rho kAv_{max}^2$$

thus: $$v_{max} = \sqrt{\frac{2(mg-\rho Vg)}{\rho kA}}$$ where $A$ is the cross section of the object and $k$ is the drag coefficient of the object for this directon.

But there are a number of caveats with this result:

• First of all, to apply Archimedes's principle, the fluid has to be in equilibrium. If the fluid is disturbed by a moving object, the principle no longer applies, and immensely complicated hydrodynamic equations and simulations can become necessary. One can hope that if the terminal velocity of the rising is small, these effects can be neglected, but this is an approximation at best.

• $k$ (also called $C_D$) in fact depends on the Reynolds number, and therefore ultimately on $v$.

• If our object is not a sphere (especially if it has planar shape or non-uniform density) it may start to tumble and rotate instead of rising with stable attitude, changing its relevant $A$ and $k$ continously.

• The density of the fluid can vary a bit if its temperature varies with depth.

• Reaching maximum (terminal) velocity is only certain if the object starts in infinite depth. If the depth and the acceleration are small, it may reaches the surface before reaching the maximum speed.

• the quadratic drag formula is only appropriate for a certain flow regime, if the velocity is very small, Stokes drag might be a more realistic solution