A basketball arcing toward the hoop, a cannonball launched from a fortress, a stream of water from a garden hose — all are examples of projectile motion. It's one of the first topics you encounter in physics, and one of the most elegant: an object moves through space under the influence of gravity alone, tracing a perfect parabolic path that's been understood mathematically since Galileo.
What Is Projectile Motion?
Projectile motion is the motion of an object launched into the air, subject only to the acceleration due to gravity (and neglecting air resistance). The key insight — the idea that makes projectile motion tractable — is that horizontal and vertical motion are completely independent.
Gravity acts only vertically, so it affects only the vertical component of motion. The horizontal component is unaffected by gravity and therefore remains constant throughout the flight (assuming no air resistance). This independence of perpendicular components is not obvious, but it is a direct consequence of Newton's second law applied component by component.
The Equations
If an object is launched with initial speed v₀ at angle θ above the horizontal, its initial velocity components are:
The horizontal and vertical positions at time t are:
And the velocity components at time t:
Notice that the horizontal velocity never changes — there's no horizontal acceleration. The vertical velocity changes linearly with time at rate g = 9.8 m/s², decreasing on the way up and increasing (in the downward direction) on the way down.
The Shape of the Path: A Parabola
If you eliminate the time variable between the x(t) and y(t) equations, you get a relationship between y and x that is a quadratic — meaning the trajectory is a parabola. This is one of Galileo's great insights: the path of a projectile is a parabolic curve, and the shape of that curve depends only on the launch angle and initial speed.
Maximum Height, Range, and Time of Flight
For a projectile launched from and returning to the same height:
Maximum height: At the top of the trajectory, the vertical velocity is zero. Setting vᵧ = 0 and solving for time gives ttop = v₀ sin θ / g. Substituting back:
Total time of flight: By symmetry, the total time is twice the time to reach maximum height:
Horizontal range: The total horizontal distance covered during the flight:
This last equation reveals something elegant: for a given launch speed, the range is maximized when sin(2θ) = 1, which occurs at θ = 45°. Launch angles equally above and below 45° produce the same range — a 30° launch goes just as far as a 60° launch (though the 60° launch goes much higher).
Common Mistakes in Projectile Motion
Mistake 1: Thinking the velocity is zero at the top. Only the vertical component of velocity is zero at maximum height. The horizontal component is still v₀ cos θ. The speed at the top is v₀ cos θ, not zero — unless the launch angle is 90° (straight up).
Mistake 2: Using the range formula for unequal launch and landing heights. The formula R = v₀² sin(2θ)/g only works when the projectile lands at the same height it was launched from. For a ball thrown off a cliff, you need to go back to the full kinematic equations and solve them directly.
Mistake 3: Forgetting that acceleration is constant. Throughout the entire trajectory — going up, at the peak, and coming down — the acceleration is g = 9.8 m/s² directed downward. It doesn't change direction or magnitude at the top. Gravity doesn't turn off at maximum height.
Why Projectile Motion Matters
Projectile motion is more than an academic exercise. It's the physics behind ballistics, sports science, civil engineering (designing bridges and arches), aerospace (rocket trajectories), and even forensic science (reconstructing crime scenes). The principles are the same at every scale: decompose the motion into independent components, apply the kinematic equations, and let the mathematics reveal the answer.
Written by
Dr. James Carter
Physicist and educator with extensive experience in classical mechanics.