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Curvilinear Motion
Two Ways to Completely Specify Particle Curvilinear Motion
Planar particle kinematics involves specifying the following: Where?
When? Direction? How fast? Is speed changing? Direction changing?
Where? This is the x-y position of a particle. (This entire discussion
also applies to r-q and other coordinate systems as well).
When? This is the time the particle is at that position.
Direction? How fast? This is velocity, the speed and direction
of the particle at that instant.
Speed changing? Direction changing? This is acceleration, the rate
at which speed and direction change.
To completely specify particle curvilinear motion, then, requires
that all of this information be supplied. In typical, practical
problems, how is this done? There are basically two ways.
Path Given: This type of problem gives the x-y path along which
the particle moves. Because the path equation alone supplies no
time information, the problem must also supply speed and acceleration
information. Usually, it will give the x or y direction velocity
and acceleration. Given two of these, and using the chain rule on
the path equation, one can calculate the remaining two.
Parametric Equations Given: This type of problem gives the x and
y parametric equations, x = f(t) and y = g(t). Given these, one
can immediately and easily calculate positions, velocity and acceleration
at any time. The only difficulty with motion specified by parametric
equations is that it can be difficult to visualize the path. For
simple problems, time can be eliminated to obtain a path equation
of y = f(x). For more difficult problems one may have to use a spreadsheet
to evaluate the parametric equations over time and plot the (x,y)
combinations to see the path.
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