In analyzing the physics of projectiles fired from ground level, one obtains the following
parametric equations:
x =cos(theta)(t)
y=sin(theta)(t -t^2)
where 0 is the angle at which the projectile is fired.
Our goal here is to obtain the set of points x, y in the plane that are reachable by our
projectile. Plot in the first quadrant the resulting parabolas as 0 ranges from 0 to /2. You
will see that a certain set of points in the first quadrant is reachable. Use calculus to find the
equation of the boundary of reachable points. (Hint: Suppose x is fixed, solve for t in terms
of x and 0 and plug into the equation for y. Then find the maximum possible value of y over
all 0 is less than or equal to theta is less than or equal to pi /2 )