Projectile motion is the motion of an object that is launched into the air and then moves primarily under the influence of gravity, with air resistance neglected or treated as a secondary effect in the simplest model. A projectile can be thrown, launched or otherwise given an initial velocity, after which gravity continuously accelerates it downward. In the idealised two dimensional model near Earth's surface, horizontal and vertical motion can be analysed independently.
If the horizontal axis is x and the vertical axis is y, and air resistance is ignored, horizontal acceleration is zero while vertical acceleration is approximately negative g, where g is about 9.81 metres per second squared near Earth's surface. The initial velocity can be separated into horizontal and vertical components. The horizontal component remains constant, while the vertical component changes uniformly because of gravity. Combining these motions produces a parabolic trajectory for an object launched at an angle and landing at the same vertical level from which it was launched. Projectile motion is therefore an important example of two dimensional motion produced by combining independent components of motion.
The standard equations for ideal projectile motion follow directly from constant acceleration. If an object is launched with initial speed v at angle theta above the horizontal, its initial horizontal velocity is v cos theta and its initial vertical velocity is v sin theta. Horizontal position after time t can be written as x equals v cos theta multiplied by t, while vertical position relative to the launch point is y equals v sin theta multiplied by t minus one half g t squared.
The vertical velocity is v sin theta minus g t, whereas the horizontal velocity remains v cos theta. These equations show that gravity does not directly alter the horizontal velocity in the ideal model. The time required to reach maximum height occurs when the vertical velocity becomes zero, giving t equal to v sin theta divided by g. If the projectile returns to the same height, the total flight time is 2v sin theta divided by g. The maximum height is v squared sin squared theta divided by 2g, and the horizontal range is v squared sin 2theta divided by g under the same ideal assumptions.
Projectile motion became historically important during the development of classical mechanics because it provided a clear example of how horizontal and vertical components can be treated separately. Galileo's investigations of falling bodies and projectiles helped establish that uniform gravitational acceleration could be analysed independently of horizontal motion. His work challenged older ideas that heavier objects necessarily fall proportionally faster and contributed to the mathematical study of motion.
The parabolic nature of ideal projectile trajectories follows from eliminating time between the horizontal and vertical position equations. Since x is proportional to time while y contains a term proportional to time squared, y becomes a quadratic function of x. Newtonian mechanics subsequently provided the force based explanation through the gravitational acceleration acting on the projectile. The historical development of projectile motion is therefore closely connected to the emergence of quantitative mechanics and the decomposition of complex motion into mathematically manageable components.
A projectile does not necessarily follow the textbook parabolic path in real life. Air resistance produces a drag force opposite to the object's motion, reducing its horizontal speed and altering its vertical motion. Because drag depends on speed and often on the square of speed, the equations become nonlinear and the trajectory is generally no longer a simple parabola. The shape also depends on the object's size, mass, shape, surface characteristics and the density of the surrounding air.
Spin can create lift through the Magnus effect, causing a ball to curve sideways or vertically. Wind changes the projectile's motion relative to the ground. For very long range projectiles, Earth's curvature and the variation of gravitational acceleration with altitude become relevant, while Earth's rotation can introduce Coriolis deflection. Consequently, the ideal projectile equations are most appropriate when the flight is relatively short, speeds are moderate and air resistance can reasonably be neglected.
Projectile motion has many practical applications. Sports such as football, basketball, cricket, baseball and athletics involve objects moving through the air under gravity and aerodynamic forces. Engineers analyse trajectories when designing throwing mechanisms, launch systems and devices that move objects through air. Fire control and ballistic calculations use more sophisticated projectile models that incorporate drag, atmospheric conditions, wind, spin, gravity variation and sometimes Earth's rotation.
Spacecraft launch vehicles initially undergo a phase of motion that can be described partly using projectile concepts, although orbital dynamics soon becomes essential because Earth's curvature and changing gravitational geometry cannot be neglected. Water fountains, streams and sprayed liquids can also display approximately projectile like trajectories when the fluid leaves a nozzle, although fluid dynamics becomes important for detailed analysis. The general principle remains that an initial velocity combined with gravitational acceleration produces a predictable path that can be calculated from the governing equations.
Projectile motion demonstrates the importance of initial conditions. Changing the launch speed changes the trajectory's scale, while changing the launch angle redistributes the initial velocity between horizontal and vertical components. For a fixed launch speed and equal launch and landing heights, the ideal range is maximised at 45 degrees, because the factor sin 2theta reaches its maximum value when 2theta equals 90 degrees.
Complementary angles such as 30 degrees and 60 degrees produce the same ideal range when all other conditions are identical, although their flight times and maximum heights differ. A lower angle produces a shorter flight time and lower maximum height, while a higher complementary angle produces a longer flight time and greater maximum height. These conclusions apply only to the ideal symmetric case. Air resistance, wind, different launch and landing elevations and terrain constraints can shift the angle that maximises practical range.
The horizontal and vertical components of projectile motion illustrate a broader principle of vector decomposition. Velocity is a vector, so it can be represented as perpendicular components that evolve according to the forces acting along each direction. Gravity acts vertically downward, so in the simplest model it changes only the vertical component. The horizontal component therefore remains constant in the absence of horizontal forces. This independence is mathematical rather than absolute.
If a horizontal force such as aerodynamic drag acts, the horizontal component also changes. The projectile equations consequently depend on the assumption that the coordinate axes are chosen conveniently and that the gravitational field is approximately uniform. The model also assumes a point like object and neglects rotation unless otherwise specified. These assumptions allow the motion to be solved analytically and make projectile motion a foundational example in introductory mechanics.
Projectile motion is particularly valuable because it connects several fundamental physical ideas: velocity, acceleration, vectors, forces, trajectories, energy and measurement. The projectile's kinetic energy changes continuously as gravitational potential energy changes, with total mechanical energy conserved in the ideal no drag case. At the highest point, the vertical velocity is zero but the horizontal velocity remains nonzero, so the projectile continues moving horizontally rather than stopping. The acceleration remains downward even at the highest point.
This is a common source of misconception. Another misconception is that an object must have an upward force acting on it while it rises. After release, if air resistance is neglected, gravity is the only force and the object rises because of its existing upward velocity while gravity reduces that velocity until it reaches zero vertically. Projectile motion therefore provides a precise demonstration that motion can continue even when acceleration is directed differently from instantaneous velocity.
