The kinematic equations describe motion in a straight line at constant acceleration, linking initial velocity, final velocity, acceleration, time, and displacement. Learn the four equations, how to choose the right one, how a velocity-time graph shows acceleration and displacement, and a worked example.
What the kinematic equations are
The kinematic equations are a set of formulas that describe motion in a straight line when acceleration is constant. They connect five quantities: initial velocity (v₀), final velocity (v), acceleration (a), time (t), and displacement (s). Knowing any three lets you solve for the rest. They rest on the definitions of position, velocity, acceleration, and time.
The four equations
v = v₀ + at
s = v₀t + ½at²
v² = v₀² + 2as
s = ½(v₀ + v)t
Each equation leaves out one of the five quantities, so you pick the equation that skips the variable you neither know nor need.
Reading a velocity–time graph
Because acceleration is constant, a graph of velocity against time is a straight line. Two features of that line carry all the physics:
On a velocity–time graph, the slope is the acceleration and the area beneath the line is the displacement.
The slope of the line equals the acceleration a, and the area under the line equals the displacement s. That area is a trapezoid, which is exactly what the equation s = ½(v₀ + v)t computes.
Worked example
A car starts at v₀ = 2 m/s and accelerates at a = 1.5 m/s² for t = 6 s. Final velocity: v = 2 + 1.5×6 = 11 m/s. Displacement: s = 2×6 + ½×1.5×6² = 12 + 27 = 39 m.