Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.

Question

a(t) = (6t + e^t) i + 12t² j,    v(0) = 3i,    r(0) = 3 i − 7 j

Answer 1

a(t) = (6t + e^t)i+12t²j

v(t) = (3t² + e^t + c₁)i+ (4t³+ c₂)j

v(0) = (1+ c₁)i +(0 + c₂)j = 3i

So, 1 + c₁ = 3 and c₂ = 0.

Therefore, c₁ = 2

v(t) = (3t² + e^t + 2)i + (4t³)j

r(t) = (t³ + e^t + 2t + c₃)i + (t⁴ + c₄)j

r(0) = (1 + c₃)I + (c₄)j = 3i + 7j

So, 1 + c

So, 1 + c₃ = 3 and c₄ = 7

Therefore, c₃ = 2

r(t) = (t³ + e^t + 2t + 2)I + (t⁴ + 7)j