Question

Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. a(t) = 2 i + 6t j + 12t2 k, v(0) = i, r(0) = 7 j − 4 k

Answer 1

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