The Kinematic Equations

Motion with constant acceleration has one underlying relationship: velocity changes steadily, and displacement is the area under the velocity-time graph. That area is a trapezoid, half the sum of the initial and final velocities times the time. From that single trapezoid area, algebra produces every other kinematic equation. v = v0 + a t comes straight from the definition of acceleration. d = v0 t + 1/2 a t² comes from splitting the trapezoid into a rectangle and a triangle. v² = v0² + 2 a d comes from combining the first two to kick time out entirely. They look like four separate tools, but they are one idea wearing different outfits, each built so a different variable disappears.

Choosing the right equation is really just an inventory problem. Write down what you know: v0, v, a, d, t, five quantities, and each equation uses only four of them. If time is missing from the problem, use v² = v0² + 2 a d, the one equation with no t in it. If displacement is missing, use v = v0 + a t. Scan for the one letter you don't have and don't need, then grab the equation that leaves it out.

These equations only work because acceleration is constant, which is exactly why this topic sits right after position, velocity and acceleration graphs. A constant-acceleration graph is a straight line on a velocity-time plot, which is what made that trapezoid area trustworthy in the first place. Once acceleration starts changing, the velocity-time graph curves, the area is no longer a simple trapezoid, and the equations quietly stop applying. That failure is exactly what leads into projectile motion, where horizontal and vertical directions each get their own constant acceleration and their own clean set of these equations.