A

The velocity of the particle is given by the equation \( \vec{v} = \begin{pmatrix} 2t \\ 3t^2 \end{pmatrix} \). We can find the position vector by integrating the velocity with respect to time.
Step 1: Integrate the x-component: \( x(t) = \int 2t \, dt = t^2 + C_x \); since the particle starts from the origin, \( C_x = 0 \), thus, \( x(t) = t^2 \).
Step 2: Integrate the y-component: \( y(t) = \int 3t^2 \, dt = t^3 + C_y \); with the same reasoning, \( C_y = 0 \), so \( y(t) = t^3 \).
Step 3: To find the equation of trajectory, we eliminate time \( t \) between the two equations: if \( x = t^2 \), then \( t = \sqrt{x} \). Substituting into the y-equation gives: \( y = t^3 = (\sqrt{x})^3 = x^{3/2} \).
Therefore, the equation of the trajectory is: \( y = x^{3/2} \).