In the previous page we found solutions for the equation of motion
which, when combined, describe the full trajectory of the plastic ball
from the moment that it is vertically thrown into the air
until it moves forever downward with a constant (terminal) velocity
and which are given by
v(t) = -vterm×tan(t/T)
for
-πT/2 < t < 0
v(t) = -vterm×tanh(t/T)
for
0 < t
With
vterm = √(g/|B|)
and
T = 1/√(g|B|)
The above equations may also be solved for the position (height)
function y=y(t).
It results in the following expressions.
y(t) = y0 + vtermT×log(cos(t/T))
for
-πT/2 < t < 0
y(t) = y0 - vtermT×log(cosh(t/T))
for
0 < t
Notice, moreover, that in the above solution for the vertical motion,
the plastic ball reaches its highest position for
t=0,
which is not completely in agreement with the experimental results.
However, that can be cured by adding a constant
t0
to t.
We will not be bothered by that unimportant detail in the following,
in order to keep the formulas simple.
Hence, in the following we have that the plastic ball
reaches its highest position for
t=0.