Integration to find velocity and position

From acceleration to velocity to position

• Integration undoes differentiation: v = ∫ a dt and s = ∫ v dt.
• Between two instants: v(t₂) − v(t₁) = ∫_t₁^t₂ a dt, and displacement = ∫_t₁^t₂ v dt.
• The two constants of integration come from the initial conditions (initial velocity and initial position at t = 0).
• When a is constant these relations reduce to the familiar equations of motion: v = v₀ + at, s = v₀t + ½at², v² = v₀² + 2as.

Worked example

1. A particle has a = 6t − 4 m/s², initial velocity 2 m/s and initial position 1 m.
   v = ∫(6t − 4)dt = 3t² − 4t + C; at t = 0, v = 2 → C = 2, so v = 3t² − 4t + 2.
   s = ∫v dt = t³ − 2t² + 2t + C′; at t = 0, s = 1 → C′ = 1, so s = t³ − 2t² + 2t + 1.
   At t = 2: v = 12 − 8 + 2 = 6 m/s and s = 8 − 8 + 4 + 1 = 5 m.