Projectile Motion

Projectile motion follows a parabolic path because horizontal displacement increases linearly with time while vertical displacement increases quadratically due to constant gravitational acceleration. When time is eliminated between the parametric equations for horizontal and vertical position, the resulting relationship between vertical and horizontal position forms a parabola, confirming the curved trajectory.

The launch angle decomposes the initial velocity vector into horizontal and vertical components. The horizontal component remains constant throughout flight because horizontal acceleration is zero. The vertical component changes continuously due to gravitational acceleration, decreasing as the projectile rises, reaching zero at the peak, then increasing downward during descent.

Projectile motion analysis assumes air resistance is negligible and gravity is the only force acting on the object. These assumptions result in constant downward vertical acceleration and zero horizontal acceleration, allowing the motion to be analyzed as two independent components: constant horizontal motion and uniformly accelerated vertical motion.

The horizontal position is linear in time, and the vertical position is quadratic in time. By solving the horizontal equation for time and substituting into the vertical equation, you eliminate the time parameter and obtain a direct relationship between vertical and horizontal position, which is the parabolic trajectory equation.

Horizontal velocity remains constant because horizontal acceleration is zero. Integrating zero acceleration yields a constant horizontal velocity component that does not change throughout the flight. This constant velocity produces linear horizontal displacement, independent of gravity's effects on vertical motion. Understanding this principle is essential for analyzing motion in space velocity and acceleration.

At the peak of the trajectory, vertical velocity becomes momentarily zero. Before reaching the peak, vertical velocity decreases due to upward motion against gravity. After the peak, vertical velocity increases in magnitude in the downward direction as the projectile falls, demonstrating the continuous effect of constant gravitational acceleration.

Starting with constant acceleration components, integrating horizontal acceleration once yields constant horizontal velocity, and integrating again gives horizontal displacement. Similarly, integrating vertical acceleration with initial vertical velocity gives vertical velocity, and integrating again yields vertical displacement. These integrals of vector functions produce the parametric equations describing position as a function of time.