Although Archimedes' principle gives the force on a buoyant object, it is generally not recognized that this does not determine the related acceleration of the object in the usual way over Newton's second law. This is because not only has the mass of the object to be accelerated but also the mass of the displaced fluid (gas). In the following, the relevant equation of motion for a buoyant object is derived. In the first part the relevant equations are derived for the case of a frictionless medium, i.e. the acceleration of the body is assumed as uniform; in the second part this result is generalized to include friction (drag), and the resultant equation of motion is solved and applied to some illustrative cases.
### Frictionless Buoyancy Acceleration

If one has an object with mass m fully submerged in a medium (fluid or gas) in hydrostatic equilibrium in a gravitational field, it will, apart from the gravitational force
_{d} is the mass of the fluid (gas) displaced by the volume of the object (Archimedes' Principle).

The total force on the object is therefore

This is indicated schematically in Fig. 1.
**Fig.1**
As an analogy, one can compare the situation to a scale, where the weight on one side is given by the object, and the weight on the other side by the displaced fluid element. Depending on which of the two is heavier, one side of the scale will drop and the other rise, but since both sides are rigidly connected, both masses have to be accelerated together at the same rate (albeit in opposite directions).

The acceleration of the object is consequently given by

It is obvious that without taking the displaced fluid element into account in the above sense, energy would not be conserved during the buoyant motion of an object as it would gain both gravitational potential energy and kinetic energy when rising in the fluid (*thanks to Christopher for addressing the issue of energy conservation in this context (which led me to look into buoyancy more closely and formulate the above theory) and also for his experimental work which indeed confirmed the weight reduction effect (Eq.(6))*).### Buoyancy Including Drag

The above consideration holds obviously only for the idealized case with no friction. In reality friction will limit its applicability to very small velocities, which in practice may be exceeded already after a fraction of s second. The correct equstion of motion must then include the friction (drag) force
_{D} the drag coefficient of the object (which is usually determined by experiment and typically has a value between 0.1-2 depending on the shape of the object).

Instead of Eq.(3), the force equation must then be written as (taking Eq.(6) into account)_{d})).

Eqs.(9) and (10) constitute the general solution for the velocity of an object in a buoyant medium. I have plotted the curve as a function of time for the case of a buoyant object in the gravitational field of the earth (g=981 cm/sec^{2} with cross section of A=1 cm^{2}, mass m=0.5g, density of the medium ρ=1g/cm^{3} (water) i.e. m_{d}=1g, and a drag coefficient C_{D}=0.47 (sphere).
**Fig.2**
The red curve is for the usual (incorrect) theory which neglects the mass of the displaced fluid when calculating the acceleration (i.e. using Eq.(4)), the blue curve does correctly take the displace fluid into account (i.e. according to Eq.(5)). Whilst the terminal velocity is identical in both cases, the usual theory substantially underestimates the time it takes to reach this velocity. The discrepancy is even more drastic for a lighter (less dense) object as shown below (here m=0.1g, where the time is underestimated by a full order of magnitude).
**Fig.3**
It is obvious from this consideration that without taking the mass of the displaced fluid into account in the equation of motion, the velocity of the buoyant object as a function of time will in general not be approximated correctly. Whilst the terminal velocity is unaffected by this, the approach time to this state is vastly underestimated in cases where the density of the object is comparable or less than the density of the fluid.

(1) F_{g} = m^{.}g

(2) F_{b} = - m_{d}^{.}g ,

The total force on the object is therefore

(3) F = F_{g}+F_{b} = (m - m_{d})^{.}g .

(4) a = F/m = g ^{.}(m - m_{d})/m .

This is indicated schematically in Fig. 1.

The acceleration of the object is consequently given by

(5) a = F/(m + m_{d}) = g ^{.}(m - m_{d})/(m + m_{d}) .

(6) ΔF = -(m - m_{d})^{.}a = -g ^{.}(m - m_{d})^{2}/(m + m_{d}) .

It is obvious that without taking the displaced fluid element into account in the above sense, energy would not be conserved during the buoyant motion of an object as it would gain both gravitational potential energy and kinetic energy when rising in the fluid (

(7) F_{D} = - 0.5^{.}ρ^{.}v^{2}^{.}C_{D}^{.}A

Instead of Eq.(3), the force equation must then be written as (taking Eq.(6) into account)

(8) F = (m + m_{d})^{.}a = (m + m_{d})^{.}dv(t)/dt =

= F_{g} +F_{b} +F_{D} = (m - m_{d})^{.}g - 0.5^{.}ρ^{.}v^{2}(t)^{.}C_{D}^{.}A.

(9) v(t) = v_{T}^{.}tanh[g^{.}(m - m_{d})/(m + m_{d}) ^{.}t/ v_{T}]

(10) v_{T} = √ {g^{.}(m_{d}-m)/0.5/ρ/C_{D}/A}

Eqs.(9) and (10) constitute the general solution for the velocity of an object in a buoyant medium. I have plotted the curve as a function of time for the case of a buoyant object in the gravitational field of the earth (g=981 cm/sec

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)

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