deriving the kinematics equations

the equations of motion apply only to objects moving with uniform (constant) acceleration.

symbols used

• u — initial velocity
• v — final velocity
• a — acceleration
• t — time
• s — displacement

derivation of v = u + at

this equation is derived directly from the definition of acceleration.

• start from the definition: a = (v − u) / t
• multiply both sides by t: at = v − u
• rearrange to make v the subject: v = u + at

derivation of s = ut + ½at²

displacement is equal to the area under a velocity–time graph.

• for uniform acceleration, average velocity is (u + v) / 2
• displacement: s = ((u + v) / 2) t
• substitute v = u + at
• s = ((u + (u + at)) / 2) t
• simplify: s = (2u + at)t / 2
• expand: s = ut + ½at²

derivation of v² = u² + 2as

this equation is obtained by eliminating time.

• start with v = u + at
• rearrange to find time: t = (v − u) / a
• substitute into s = ((u + v) / 2) t
• s = ((u + v) / 2) × ((v − u) / a)
• use difference of squares: (u + v)(v − u) = v² − u²
• s = (v² − u²) / (2a)
• rearrange: v² = u² + 2as