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Viscous media

Resistance against motion

For an object that moves vertically in a fluid, the resistance is usually a combination of a linear and a quadratic term, in the velocity v of the object with respect to the fluid (A and B are supposed to be constants):

a = -g + A×v + B×v²

However, in air, under the conditions of our experiments, the linear term is completely negligible.
For the solid metal ball even B is too small to be observed.
But, for the plastic ball we measured to some accuracy the value of B.
If you have made a graph for the velocity, then you could observe that on coming downward the speed of the plastic ball increases when time evolves. This is normal for a falling object. However, in vacuum, where resistance is absent (A_vacuum = B_vacuum = 0), the increase is constant in equal intervals of time and equal to the acceleration g of gravity. But, for the plastic ball the speed increases less and less. Indeed, if we had not stopped the experiment at the instant the plastic ball returned to us, for example by performing the experiment at a huge tower such that the ball can continue to fall after it passed us, then we would find that at a certain speed the increase in speed vanishes. The ball can simply not fall faster in air. The gravitational force, which pulls the plastic ball towards the Earth, is exactly balanced by the force of the air resistance, which works in the upward direction, opposite to the downward direction of the velocity.
Let us designate by v_term the terminal speed of the falling plastic ball. Eventually, also the metal ball will reach its terminal speed. But that is at much larger values for its speed.
Once one knows the value of B one may deduce the terminal speed for the plastic ball, because no increase in speed means vanishing acceleration:

0 = -g + B×v_term²

which gives us for the terminal speed the relation

v_term = √(g/B)

Hence from the average value 0.025 m^-1 which we obtained before for B, we obtain that the terminal speed of the freely falling plastic ball in air equals 20 m/s. This value corresponds well to the value one obtains using a graphical representation of the data.
Notice that the above formula also serves for an experimental determination of the value of B for fluids once one knows that the conditions for a quadratic relation are satisfied. Just measure the terminal speed for the object when it falls downward and determine

B = g/v_term²

Such method is actually used, but then for viscous liquids which are very sticky and for which the linear term A dominates (B≅0) and in which metal bearing balls, which are first collocated at the surface of the liquid, take some time to start falling due to the surface tension and then reach very fast the terminal velocity.

the energy balance