Answer:
v(t) = (4t² + 1)î + (4t² - 1)j + (t + 1)k
r(t) = [(4t³/3) + t + 1]î + [(4t³/3) - t]j + [(t²/2) + t]
Step-by-step explanation:
r(0) = ‹1, 0, 0› = î
v(0) = î - j + k
a(t) = 8ti + 8tj + k
To find the velocity vector at any time t
Since a(t) = v'(t),
a(t) = (dv/dt)
a(t) = (dv/dt) = 8ti + 8tj + k
(dv/dt) = 8ti + 8tj + k
dv = (8ti + 8tj + k)dt
∫ dv = ∫ (8ti + 8tj + k)dt
Integrating the left hand side from from v(0) to v(t) and the right hand side from 0 to t
v - v(0) = (4t²î + 4t²j + tk)
v(t) = v(0) + (4t²î + 4t²j + tk) = (î - j + k) + ((4t²î + 4t²j + tk))
v(t) = (4t² + 1)î + (4t² - 1)j + (t + 1)k
For its position vector at any time
v(t) = r'(t)
v(t) = (dr/dt) = (4t² + 1)î + (4t² - 1)j + (t + 1)k
(dr/dt) = (4t² + 1)î + (4t² - 1)j + (t + 1)k
dr = [(4t² + 1)î + (4t² - 1)j + (t + 1)k] dt
∫ dr = ∫ [(4t² + 1)î + (4t² - 1)j + (t + 1)k] dt
Integrating the left hand side r(0) to r(t) and the right hand side from 0 to t
r - r(0) = [(4t³/3) + t]î + [(4t³/3) - t]j + [(t²/2) + t]
r(t) = r(0) + [(4t³/3) + t]î + [(4t³/3) - t]j + [(t²/2) + t]
r(t) = î + [(4t³/3) + t]î + [(4t³/3) - t]j + [(t²/2) + t]
r(t) = [(4t³/3) + t + 1]î + [(4t³/3) - t]j + [(t²/2) + t]