Projectile Motion Theory

Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration due to gravity. The object is called a projectile, and its path is called its trajectory.

Key Equations

The motion can be analyzed by separating it into horizontal and vertical components:

• Horizontal position: $x(t) = v_0 \cos(\theta) \cdot t$
• Vertical position: $y(t) = h_0 + v_0 \sin(\theta) \cdot t - \frac{1}{2}gt^2$
• Trajectory equation: $y = h_0 + xtan(\theta) - \frac{gx^2}{2v_0^2\cos^2(\theta)}$

Important Parameters

Time of Flight: The total time the projectile remains in the air. Found by solving $y(t) = 0$:

$t_{flight} = \frac{v_0\sin(\theta) + \sqrt{v_0^2\sin^2(\theta) + 2gh_0}}{g}$

Maximum Height: The highest point reached by the projectile. Occurs when vertical velocity equals zero:

$h_{max} = h_0 + \frac{v_0^2\sin^2(\theta)}{2g}$

Horizontal Range: The horizontal distance traveled:

$R = v_0\cos(\theta) \cdot t_{flight}$

Assumptions

• Air resistance is neglected
• Gravitational acceleration is constant ($g = 9.81$ m/s²)
• The Earth's curvature is negligible for the distances involved
• The projectile is treated as a point mass

Applications

Understanding projectile motion is crucial in many fields:

• Sports: Basketball shots, golf drives, soccer kicks
• Military: Artillery and ballistics calculations
• Engineering: Water fountain design, fireworks displays
• Physics Education: Demonstrating kinematic principles