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Dynamics
Equation of motion
The expression which we wrote
at one
of the previous pages, namely
a = -g + A×v + B×v2
is the equation of motion for an object that falls vertically
in a viscous medium.
The equation can be turned into a
differential
equation,
since the acceleration is the first time
derivative
of the velocity:
a = dv/dt
We obtain then for the equation of motion the following expression.
dv/dt = -g + A×v + B×v2
In this expression
g,
A
and B
are supposed to be known and not dependent on the instant of time
(constant in time, or just constant).
Consequently, the only unknown is the velocity
which is a function of time
v=v(t).
How do we solve
such differential equation?
In general there is no method for
solving differential
equations.
Those which have been solved in the past centuries
can be found in
books,
or in
tables.
Others are approached by
numerical
methods,
which allow to give some insight in the structure
of their
possible
solutions.
Here, we will study three cases:
A=B=0