Answer:
v(t) = (2t + 1)i + 3t²j + 4t³k
r(t) = (t² + t)i + (t³ + 7)j + (t⁴ - 4)k
Explanation:
a(t) = 2i + 6tj + 12t²k
v(t) = ∫a(t)dt
= ∫(2i + 6tj + 12t²k)dt
= 2ti + (6t²/2)j + (12t³/3)k + c
= 2ti + 3t²j + 4t³k + c
v(0) = i
i = 0i + 0j + 0k + c
c = i
∴ v(t) = 2ti + 3t²j + 4t³k + i
v(t) = (2t + 1)i + 3t²j + 4t³k
r(t) = ∫ v(t)dt
= i ∫ (2t + 1)dt + 3j ∫ t²dt + 4k ∫ t³dt
= i (2t²/2 + t) + 3j(t³/3) + 4k(t⁴/4) + d
= i (t² + t) + jt³ + t⁴k + d
r(0) = 7j - 4k
0i + 0j + 0k + d = 7j - 4k
d = 7j - 4k
∴ r(t) = (t² + t)i + t³j + t⁴k + 7j - 4k
r(t) = (t² + t)i + (t³ + 7)j + (t⁴ - 4)k
