Kinematics: What It Is and the Equations of Motion

What kinematics studies

According to OpenStax’s University Physics Volume 1, kinematics is the branch of physics that tracks motion through four quantities — position, time, velocity, and acceleration — while the forces that produce that motion belong to dynamics, the natural counterpart covered in the Recap on Newton’s three laws of motion. The textbook starts with the simplest case, motion along a straight line, before moving on to motion in two or three dimensions.

The Treccani Encyclopedia defines kinematics as the part of mechanics that tracks over time how the geometric features of moving bodies change. For the encyclopedia, motion is a relative concept: saying a body moves only makes sense when comparing it with another body taken as a reference.

Position and frame of reference

The position of an object is the point where it is at one precise instant. Fixing it means choosing a frame of reference: a set of measuring axes, the same coordinate language used by the coordinate plane. Displacement looks only at the difference between two positions; distance traveled instead adds up the length of the entire path, and the two quantities match only when the path is a straight segment in one direction.

The curve a moving point traces as it travels is its trajectory, in the Treccani Encyclopedia’s definition.

Average velocity, instantaneous velocity, and types of motion

Average velocity is the ratio between total displacement and the time it took to cover it. Instantaneous velocity is that same ratio calculated over a time interval that keeps shrinking, nearly to zero. In uniform motion, says the Treccani Encyclopedia, the ratio between distance covered and time taken stays constant: that constant is the velocity of the uniform motion (a future Recap on uniform rectilinear motion will cover the formula and worked problems for this case). In non-uniform motion, instead, the velocity at an instant equals the derivative of distance with respect to time at that same instant — a more technical way of writing the same idea of instantaneous velocity that OpenStax explains in words.

Reading motion on a graph

On a position-time graph, the slope of the curve at a point tells you the instantaneous velocity at that instant: a steeper curve describes a faster body. That same pattern shows up again on the velocity-time graph, where the slope of the curve tells you the instantaneous acceleration. Acceleration, in general, measures how quickly velocity changes; the Treccani Encyclopedia formalizes it as the second derivative of position with respect to time.

Free fall: the classic case of constant acceleration

Practical example: A stone dropped from a bridge is in free fall, because its motion is caused only by gravity, as NASA’s Glenn Research Center defines it. Starting from zero initial velocity, after a time t the velocity reached equals the gravitational acceleration multiplied by the time elapsed, and the distance traveled equals half the acceleration multiplied by the square of the time. The stone’s weight, size, and shape don’t change the outcome: with no air resistance, every object receives the same acceleration.

Gravitational acceleration stays constant throughout the fall: that’s why velocity grows in direct proportion to time while the distance traveled grows with the square of time, the reason the velocity-time graph of a free fall is a straight line and the position-time graph is a parabola. On Earth, gravitational acceleration ranges from place to place between 9.78 and 9.83 m/s²; school calculations use the rounded average value, 9.8 m/s².

Uniform circular motion

Uniform circular motion is the motion of an object traveling around a circle while keeping its speed constant. At constant speed, though, that object still accelerates: the speed doesn’t change, but the direction does, continuously, and this acceleration, called centripetal, has a magnitude equal to the square of the velocity divided by the radius, pointed toward the center of the circle. Observing and measuring these motion patterns follows the same process described in the Recap on the scientific method: the same logic of position, velocity, and acceleration also applies to the motion of the solar system’s planets, even though their orbits aren’t circular.