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Velocity and acceleration of a particle

Short answer

Integrate acceleration twice with initial conditions v(0)=0, r(0)=0. v = 6 sin 2t i + 4(cos 2t − 1) j + 8t² k, r = 3(1 − cos 2t) i + (2 sin 2t − 4t) j + (8t³/3) k.

The acceleration of a moving particle at any time t is given by d²r/dt² = 12 cos 2t i − 8 sin 2t j + 16t k. Find the velocity v and displacement r at any time t if t = 0, v = 0 and r = 0.

Given d²r/dt² = 12cos 2t i − 8sin 2t j + 16t k, with v = 0 and r = 0 at t = 0. Integrate twice and use the initial conditions.

Velocity

v = dr/dt = ∫ (12cos 2t i − 8sin 2t j + 16t k) dt
  = 6 sin 2t i + 4 cos 2t j + 8t² k + c₁
At t = 0, v = 0:  0 + 4 j + 0 + c₁ = 0  → c₁ = −4 j
v = 6 sin 2t i + (4 cos 2t − 4) j + 8t² k

Displacement

r = ∫ v dt
  = −3 cos 2t i + (2 sin 2t − 4t) j + (8t³/3) k + c₂
At t = 0, r = 0:  −3 i + c₂ = 0  → c₂ = 3 i
r = 3(1 − cos 2t) i + (2 sin 2t − 4t) j + (8t³/3) k

Answer:

  • v = 6 sin 2t i + 4(cos 2t − 1) j + 8t² k
  • r = 3(1 − cos 2t) i + (2 sin 2t − 4t) j + (8t³/3) k

✓ Answer checked by Claude Opus 5.5 (high effort)

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