Throw a ball and watch its arc. That parabolic path is projectile motion, one of the most fundamental topics in introductory physics. The key insight is that horizontal and vertical motion are independent. The horizontal velocity stays constant (ignoring air resistance), while the vertical velocity changes due to gravity. This separation makes the math manageable and the predictions accurate.
The basic equations come from kinematics. Horizontal distance equals horizontal velocity times time. Vertical position equals initial height plus vertical velocity times time minus one half g times time squared. The vertical velocity at any point is the initial vertical velocity minus g times time. With these four equations, you can solve for anything: maximum height, range, time of flight, velocity at impact, and more.
Maximum range occurs at a 45-degree launch angle, but only on flat ground with no air resistance. In reality, air resistance shifts the optimal angle lower, typically to about 40 degrees for a baseball. Golf ball drivers optimize for even lower angles because backspin creates lift. Our Projectile Motion Calculator handles the basic cases without air resistance.
Maximum height depends only on the vertical component of velocity. Launch at 30 m/s at 90 degrees, and you reach about 46 meters. Launch at the same speed at 45 degrees, and you only reach about 11.5 meters vertically but travel much farther horizontally. The trade-off between height and range is fundamental to projectile design.
Time of flight is determined by how long it takes the projectile to go up and come back down. For a launch from ground level, this is 2v sine theta divided by g. A ball thrown at 20 m/s at 60 degrees stays airborne for about 3.5 seconds. That same ball at 30 degrees only stays up for about 2 seconds but covers more ground.
Real-world projectiles are more complicated. Air resistance creates drag proportional to velocity squared, making the equations differential rather than algebraic. Baseballs, with their seams and spin, experience the Magnus effect, which curves their path. Artillery shells travel fast enough that the curvature of Earth and Coriolis effect matter. These factors turn simple projectile motion into computational fluid dynamics problems.
Despite the complications, the basic model is remarkably useful. It works well for dense objects at moderate speeds. Basketballs, cannonballs, and water fountains all follow roughly parabolic paths. Engineers use it to design water slides, ski jumps, and even to calculate how far debris will scatter in an explosion.