Projectile motion combines constant horizontal velocity with accelerating vertical motion due to gravity. This creates parabolic trajectories for objects launched horizontally or at an angle, like cannonballs or javelins.

Calculations for projectile motion involve range, time of flight, and maximum height. These use trigonometry and kinematic equations to analyze the separate horizontal and vertical components of motion.

Projectile Motion Fundamentals

Motion of projectiles

• Projectile motion components split into horizontal motion maintains constant velocity while vertical motion accelerates due to gravity
• Horizontal launch begins with zero initial vertical velocity creates parabolic trajectory (cannonball fired from cliff)
• Angular launch involves initial velocity with both horizontal and vertical components forms parabolic path (javelin throw)
• Vector analysis resolves initial velocity into x and y components using trigonometric functions (sin, cos, tan)

Projectile trajectory calculations

• Range (R) measures horizontal distance traveled calculated by $R = v_0 \cos(\theta) t$ (football field goal kick)
• Time of flight (t) represents total airborne duration for angular launch found using $t = \frac{2v_0 \sin(\theta)}{g}$ (baseball pitch)
• Maximum height (h) indicates highest point reached by projectile determined by $h = \frac{v_0^2 \sin^2(\theta)}{2g}$ (fireworks display)
• Kinematic equations apply separately to horizontal and vertical motions solve for unknown variables (rocket launch)

Advanced Projectile Concepts

Initial velocity for target hits

• Target analysis considers position coordinates (x, y) and launch angle (archery)
• Reverse calculation uses range and height formulas to solve for initial velocity may require quadratic equation (golf drive)
• Minimum velocity concept determines smallest initial speed needed to reach target often involves optimization techniques (basketball free throw)

Independence of projectile motions

• Principle of independence allows separate analysis of x and y components as they do not affect each other (shotput throw)
• Constant horizontal velocity with no acceleration in x-direction distance traveled horizontally proportional to time (skipping stone)
• Accelerated vertical motion due to constant gravity acceleration velocity changes in y-direction (water fountain)
• Superposition combines horizontal and vertical motions to describe full parabolic trajectory (long jump)