Parameterised Projectile Motion

In some situations, we may have some control over the initial conditions of a projectile motion trajectory. If this is the case, we may want to select these initial conditions to optimise some property like the altitude at some location, or the range of the projectile.

For example, we may not be able to control the launch speed, but we might have control over the launch angle. Suppose we were solving the following problem, related to the previous example of a soccer ball being kicked

A soccer player is taking a penalty shot at a distance of 11 metres from the goal posts. When they kick the ball, it leaves the ground at a speed of 20 metres per second. At what angles to the horizontal can they kick the ball so that it passes between the goal posts and under the cross-bar, which is 2.44m above the ground. Acceleration due to gravity is 9.8 m/s² downwards.

To solve this problem, we can parameterise the projectile motion of the ball using the launch angle $\theta$ . We want to find all angles such that the following conditions are met:

The range of the ball's motion is at least 11 metres
The height of the ball is less than 2.44 metres at a horizontal distance of 11 metres from the launch position

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