Describe the path of a football that has been punted. at what angle should the ball be kicked to get maximum height?

a) The path is usually modeled as a parabola. (In real life, it is most definitely not a parabola, as air resistance and "flight" are definite factors.)

b) To get maximum height for a given speed, the ball should be launched straight up. (In real life, the kicker's effective speed will vary with launch angle.)

Step-by-step explanation:

The equations modeling air resistance and/or the fluid dynmics of an odd shape through air are impossible to solve "by hand", so we use equations that we can solve. For many purposes (teaching equations, functions, and solution processes, for example), the equation of a parabola is "good enough."

Vertical height is determined by vertical speed. For a given launch angle, the vertical speed is the launch speed multiplied by the sine of the angle (relative to the ground). The sine is a maximum when the angle is 90°, or straight up.