When a projectile moves near the surface of the
earth, the acceleration can be taken as constant, directed down, and equal
to the free fall accelearation of any object. (We ignore wind resistance.)
The free fall acceleration is taken to have magnitude, g. The direction of
the acceleration is down; the negative y direction for the coordinate
system shown in the figure.
Taking x to be horizontal and y to be vertical, the equations for position components are shown in the figure. Taking the origin to be at the initial point of the trajectory, we can easily eliminate time from the two position equations. The result is an equation for the trajectory; the equation that gives the vertical displacment (Y) as a function of horizontal displacement (X).
We can think of this equation as having the x and y components of initial velocity (V sub 0) as parameters. Alternatively, we can write the parameters in terms of the initial velocity magnitude and direction. Both parameterizations are shown in the figure.
Forward to circular motion.
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