Projectile motion is the motion of an object thrown into the air under the influence of gravity alone. Common examples include a ball thrown at an angle, an arrow shot from a bow, or water from a fountain. We make key assumptions: no air resistance, uniform gravitational field, and motion in a vertical plane. The fundamental concept is that projectile motion combines horizontal uniform motion with vertical uniformly accelerated motion due to gravity.
In projectile motion, we break down the initial velocity into horizontal and vertical components. 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. The horizontal velocity remains constant throughout the flight because there's no horizontal acceleration. However, the vertical velocity changes continuously due to gravitational acceleration. This vector decomposition allows us to analyze the two motions independently, making projectile motion problems much easier to solve.
The equations of motion for projectile motion are derived from basic kinematics. For horizontal motion, the position is x equals v-zero cosine theta times t, and the horizontal velocity remains constant at v-x equals v-zero cosine theta. For vertical motion, the position follows y equals v-zero sine theta times t minus one-half g t squared, while the vertical velocity changes according to v-y equals v-zero sine theta minus g t. These equations allow us to predict the projectile's position and velocity at any time during its flight.
The trajectory equation is derived by eliminating time from the parametric equations. We start with t equals x divided by v-zero cosine theta, then substitute this into the y equation to get y equals x tan theta minus g x squared over two v-zero squared cosine squared theta. This equation shows that all projectile paths are parabolic. Different launch angles with the same initial speed create a family of parabolic curves, each with its own unique shape and range.
The key parameters of projectile motion are range, maximum height, and time of flight. The range formula is R equals v-zero squared sine two theta over g. Maximum height is H equals v-zero squared sine squared theta over two g. Time of flight is T equals two v-zero sine theta over g. The maximum range occurs at 45 degrees. Complementary angles like 30 and 60 degrees give the same range. This interactive demonstration shows how these parameters change with launch angle.