O-Level Differentiation and integration: worked solution

Question

With v = 3t² - 8t + 4, work out the total path length covered by the particle over the first 3 seconds.

Find when v = 0: 3t² - 8t + 4 = 0 → (3t - 2)(t - 2) = 0 → t = 2/3 or t = 2. Displacement s = ∫v dt = t³ - 4t² + 4t. s(0) = 0 s(2/3) = 8/27 - 16/9 + 8/3 = 32/27 s(2) = 8 - 16 + 8 = 0 s(3) = 27 - 36 + 12 = 3 Distances over each interval where direction is constant: [0, 2/3]: |32/27 - 0| = 32/27 [2/3, 2]: |0 - 32/27| = 32/27 [2, 3]: |3 - 0| = 3 Total distance = 32/27 + 32/27 + 3 = 64/27 + 3 = 145/27 ≈ 5.37 cm.