Use the fundamental theorem of calculus to find the velocity and position functions for the particle. We're given
a (t ) = 3t i + 2 j
v (0) = i - 4 j
x (0) = 4 i + 2 j
Then the velocity at time t is





and the position at time t is





(a) As shown above,
v (t ) = (3/2 t ² + 1) i + (2t - 4) j
(b) Also as shown,
x (t ) = (1/2 t ³ + t + 4) i + (t ² - 4t + 2) j
(c) First, find the particle's position at t = 3 :
x (3) = (1/2•3³ + 3 + 4) i + (3² - 4•3 + 2) j = 41/2 i - j
The particle's distanc from the origin is the magnitude of this vector:
|| x (3) || = √((41/2)² + (-1)²) = √1685/2 ≈ 20.52