Derive the First Equation of Motion (Calculus & Graphical Methods)

The first equation of motion connects an object’s final velocity (v), initial velocity (u), constant acceleration (a), and elapsed time (t). Under constant acceleration, velocity changes uniformly with time. Using calculus, start from the definition of acceleration as the time rate of change of velocity: a = dv/dt. Separating variables and integrating from u → v over the interval 0 → t gives:

∫(u→v) dv = ∫(0→t) a dt ⟹ v − u = a t ⟹ v = u + a t

This relation is valid for any motion with constant acceleration (e.g., free fall ignoring air resistance, cars accelerating at a fixed rate, conveyor mechanisms). Conceptually, it states that final velocity equals initial velocity plus the uniform increment added each second.

Graphical Method (v–t Diagram)

In a velocity–time graph, constant acceleration is represented by a straight line with slope slope = (v − u)/t = a. Rearranging yields the same result: v = u + a t. 

The linearity shows why the formula is so robust acceleration is the uniform rate at which velocity increases every unit time. 

For content planning or lab demonstrations, pair this derivation with a simple data-logging setup (motion sensor or phone accelerometer) to produce a v–t plot that visually confirms the straight-line relationship and measured slope equal to a.