The terminal velocity is the highest velocity gained by an object which is falling through a fluid. The acceleration of the object will be zero as the net force acting on it will be zero. Terminal velocity is useful for many applications like studying precipitation in clouds, falling of fruits to develop separator machines, etc. In this article, we will discuss the derivation of terminal velocity with some real-life examples.
The formula of terminal velocity
Mathematically, it is represented as
Where,
vt is given as the terminal velocity,
m is given as the mass of the falling object,
g, the acceleration due to gravity,
Cd is the drag coefficient,
𝜌 density of the fluid through which the object is falling
A is the projected area of the object
Derivation of terminal velocity
According to the drag equation,
F = bv ²
As b is the constant. The value of the constant b is different for different drags.
The freefall of an object is given as,
Σ F = ma
mg -bv ² = m dv/dt
¹/m dr = dv / mg – bv²
Integrating them,
Where,
dv = α sec h² ( Φ ) d Φ
v² = α tan h² (Φ)
After integrating,
v(t) = α tanh ( αb /m × t + arc tanh ( v 0 / α )
By substituting,
After substituting v t,
Research Articles
In this article titled “Modeling of Sapodilla Fruit (Manilkara zapota (L.) van Royen) Terminal Velocity in Water,” the authors aim to determine the terminal velocity of sapodilla in order to develop a sorter machine. Terminal velocity is a property that is used widely in the development of sorting equipment for fruits. In this article, you can see the mathematical modeling for the derivation of the terminal velocity. The terminal velocity was also verified by experimentally dropping the fruit.
You can read this article titled Can terminal settling velocity and drag of natural particles in the water ever be predicted accurately? to know about the accuracy of the measurements in water. In water natural particles(not spheres) behave a bit differently from round particles. To accurately estimate the terminal velocity in a liquid setting one has to one needs to consider a force balance in which the drag force balances the difference between buoyancy and weight. So, in this article, the authors derive the formula for terminal velocity.
In this article titled Terminal Velocity of Single Drops in Stagnant Liquids, the authors determine the terminal velocities of single drops rising through infinite stagnant liquids.
In this article titled “Vertical Structure of Ice Clouds and Vertical Air Motion from Vertically Pointing Cloud Radar Measurements,” the authors aim to study the vertical structure of ice clouds and vertical air motion. The terminal fall velocity (Vt) of a hydrometeor is the most important parameter in ice cloud parameterization, and it strongly influences the sedimentation of ice crystals in numerical models. So the authors calculate the terminal fall velocity and find the relationship between the terminal fall velocity and the distributions of reflectivity(cloud parameter).
In this article titled “Terminal settling velocity of solids in the pseudoplastic non-Newtonian liquid system – Experiment and ANN modeling” the terminal settling velocity of solid particles is experimentally determined in fluidizing columns. The authors have made an empirical correlation for deriving the terminal settling velocity with acceptable statistical accuracy.
What are the real-life applications of terminal velocity?
Terminal velocity is useful in developing weapons, manufacturing/designing separator machines, fluid mechanics, studying cloud formation, etc.
Which scientist discovered terminal velocity?
Galileo discovered terminal velocity.
