Welcome to projectile motion! This is one of the most fundamental topics in physics. Projectile motion describes how objects move through the air when they're thrown, launched, or dropped. The key insight is that gravity is the only force acting on the projectile, assuming we ignore air resistance.
To analyze projectile motion, we need to break the initial velocity into components. The initial velocity vector v-zero can be decomposed into horizontal and vertical components using trigonometry. The horizontal component is v-zero-x equals v-zero cosine theta, and the vertical component is v-zero-y equals v-zero sine theta, where theta is the launch angle above the horizontal.
Now let's analyze the motion in each direction separately. Horizontally, there's no acceleration, so the projectile moves with constant velocity. The horizontal position is simply x equals v-zero-x times t. Vertically, gravity causes constant downward acceleration. The vertical position follows y equals v-zero-y times t minus one-half g t squared, and the vertical velocity is v-y equals v-zero-y minus g t.
There are three key parameters we can calculate for projectile motion. The time of flight is how long the projectile stays in the air, given by two v-zero-y over g. The maximum height occurs at the peak of the trajectory and equals v-zero-y squared over two g. The range is the horizontal distance traveled, which is v-zero squared sine two theta over g. Notice that maximum range occurs at 45 degrees.
Let's compare different launch angles to see how they affect the trajectory. At 15 degrees, we get a low, short path. At 30 degrees, the trajectory is higher with more range. At 45 degrees, we achieve maximum range - this is the optimal angle for distance. At 60 degrees, the projectile goes higher but doesn't travel as far horizontally. At 75 degrees, we get a very high but short trajectory. Understanding these relationships helps us choose the right angle for different applications.