Motion in two or three dimensions

Motion in two or three dimensions is analyzed by separating the motion into independent perpendicular components. Each component obeys the same one-dimensional kinematic equations, with its own initial conditions and acceleration.

The key insight is that what happens along one axis does not affect what happens along a perpendicular axis — a horizontal force changes only the horizontal velocity, and a vertical force changes only the vertical velocity.

The velocity and acceleration vectors are found by differentiating the position components with respect to time, then combining the resulting components using the Pythagorean theorem (for magnitude) and the inverse tangent (for direction).

Projectile motion

Projectile motion is the most important special case of two-dimensional motion, in which an object moves under the influence of gravity alone, with air resistance neglected.

• The horizontal acceleration is zero, so the horizontal velocity remains constant throughout the flight.
• The vertical acceleration is g directed downward, so the vertical velocity changes uniformly.

Horizontal: x = x₀ + v₀ₓt, vₓ = v₀ₓ (constant) Vertical: y = y₀ + v₀ᵧt − ½gt², vᵧ = v₀ᵧ − gt

Solving any projectile problem requires resolving the initial velocity into horizontal and vertical components, using the vertical equation to find the time of flight, and substituting that time into the horizontal equation to find the range or horizontal position.

The independence of dimensions explains why an object launched horizontally and an object dropped vertically from the same height always hit the ground at the same instant, regardless of horizontal speed.