$$\rightleftharpoonup{xx}$$
$$\longleftharp{xx}$$,
$$\longrightharp{xx}$$,
Projectile motion models the flight of an object launched into the air, such as a soccer ball kicked during a penalty, under the simplifying assumption that air resistance is negligible. When gravity is the only force, the object experiences a steady downward acceleration at all times. This single fact explains why projectile motion can be analyzed as two independent motions happening simultaneously: a horizontal motion that does not speed up or slow down, and a vertical motion that continually changes because of gravity.
The initial velocity of the projectile is treated as a vector pointing in the launch direction. Using the launch angle, this initial velocity is separated into a horizontal component and a vertical component. Gravity affects only the vertical component. As a result, the horizontal acceleration is zero, so the horizontal velocity remains constant throughout the flight. The horizontal displacement, therefore, increases at a constant rate, producing a straight-line relationship with time.
In contrast, the vertical acceleration is constant and downward because of gravity. The vertical velocity decreases steadily as the projectile rises, becomes momentarily zero at the peak, and then increases in magnitude in the downward direction as the projectile falls. Because the vertical velocity changes at a constant rate, the vertical displacement follows a curved, time-squared pattern rather than a straight-line pattern.
The motion is naturally described with vector-valued parametric equations by expressing the position as a function of time: one time-based expression for horizontal position and another for vertical position. The horizontal position linearly depends on time, while the vertical position depends on time in a quadratic way due to the constant acceleration of gravity. When time is eliminated between these two relationships, the remaining connection between vertical position and horizontal position is a parabola. This is why the trajectory of an ideal projectile, launched and landing under uniform gravity with negligible air resistance, forms a parabolic arc.
