Particle Motion — Complete Analysis
Given v(t), find everything:
When does the particle change direction? When v(t) = 0 AND changes sign.
Total distance from t=0 to t=5: ∫ from 0 to 5 |v(t)| dt. Split at zeros of v(t) and take absolute values.
Displacement from t=0 to t=5: ∫ from 0 to 5 v(t) dt = s(5) - s(0). Can be negative.
Position at t=5 given s(0)=2: s(5) = s(0) + ∫ from 0 to 5 v(t) dt = 2 + displacement.
When is the particle farthest from origin? Check critical points (where v=0) and endpoints. Evaluate s(t) at each.
Speeding up vs slowing down:
Speeding up: v and a same sign
Slowing down: v and a opposite signs
Reference:
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