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.