Projectile Motion: Equations

Projectile motion is analyzed by treating horizontal and vertical components independently. Set the vertical acceleration to negative g and horizontal acceleration to zero. Resolve initial velocity into components using trigonometric relations. This separation simplifies the motion equations, allowing you to solve for time, displacement, and velocity in each direction separately, with time as the common variable between them.

At the zenith, or highest point of the trajectory, the vertical velocity component becomes zero. By substituting this zero value into the velocity equation for the y-direction, you can calculate the time for half-flight. This relationship is fundamental for determining both the maximum height and the total flight duration of the projectile.

The zenith height is found by substituting the half-flight time expression into the displacement equation for the y-direction. Half-flight time is calculated from the condition that vertical velocity equals zero at the peak. This yields the maximum height the projectile reaches above its launch point during flight.

The total duration of flight is exactly twice the time of half-flight. Since projectile motion is symmetric about the zenith, the time to reach maximum height equals the time to fall back to the launch level. Substituting total flight time into the horizontal displacement equation yields the maximum range equation.

The trajectory equation is derived by rearranging the horizontal position equation and substituting it into the vertical position equation. This mathematical manipulation eliminates time and produces a relationship between vertical and horizontal positions. The resulting equation has the form of a quadratic function, which geometrically represents a parabola in two-dimensional space.

Initial velocity components are obtained using trigonometric relations. The horizontal component equals v·cos(θ) and the vertical component equals v·sin(θ), where v is the velocity magnitude and θ is its direction relative to the horizontal. These components are then used in the simplified motion equations for each direction.

Time is the only common variable between horizontal and vertical motions. While horizontal and vertical displacements are analyzed independently using separate kinematic equations, time links both components together. By solving for time in one direction, you can use that same value to find displacement or velocity in the other direction.