A particle starts from rest and accelerates uniformly at

Solution

We are given a three-part motion: Uniform acceleration from rest for 5 seconds, with a = 2 m/s² Uniform motion at constant speed for 5 seconds Uniform deceleration to rest in 2 seconds We are to find the total displacement. Displacement during acceleration Initial velocity u = 0, acceleration a = 2 m/s², time t = 5 s Using: s₁ = ut + ½at² = 0 + ½(2)(5²) = 25 m Final velocity after this phase: v = u + at = 0 + 2×5 = 10 m/s Displacement during constant velocity Now the particle moves at v = 10 m/s for 5 s: s₂ = v × t = 10 × 5 = 50 m Displacement during deceleration In this phase, the particle slows from 10 m/s to 0 in 2 s Use: s₃ = ut + ½at² = 10×2 + ½(−5)(2²) But better: since deceleration is uniform: Use: s₃ = average speed × time = (10 + 0)/2 × 2 = 10 m Total displacement: s = s₁ + s₂ + s₃ = 25 + 50 + 10 = 85 m