Motion when acceleration is function of time

Motion when a is a given function of time

If a = f(t), velocity and position are found by successive integration, using the initial conditions.

1. Velocity: a = dv/dt, so dv = f(t) dt, and v = v₀ + ∫₀ᵗ f(t) dt
2. Position: v = dx/dt, so x = x₀ + ∫₀ᵗ v dt
3. The constants v₀ and x₀ are the velocity and position at t = 0 (or any known instant).

Example: a = 6t m/s² with v₀ = 2 m/s and x₀ = 0:

• v = 2 + ∫6t dt = 2 + 3t²
• x = ∫(2 + 3t²) dt = 2t + t³

At t = 2 s: v = 14 m/s and x = 12 m.

If the particle changes direction (v passes through zero), the total distance travelled is found by adding the distances in each interval separately.