How do you find the initial velocity of a projectile given the height of the launch site and distance from the base of the launch site to the projectile's landing spot?

The way to understand projectiles is to treat their speeds as two different speeds. Treat one speed as how fast the projectile moves up or down relative to the ground, and treat the other speed as how fast the object moves left to right. This is called splitting the velocity into components. Each component is independent of the other, meaning that no matter what happens to the x (the horizontal speed) component, the y (the vertical component) will be unaffected. To determine these components, you will need to know how long the object was in the air. Take the total x component and divide it by the time the object is in the air. That is the x component. Then take that and multiply it by the inverse cosine function of whatever angle the projectile was launched at. That will give you the inital velocity of the object. --An AP Physics Student Bored in Study Hall