Given,
Velocity in $x$-direction is, $\frac{dx}{dt}=v_x=2 \mathrm{m} {\mathrm{s}}^{-1}$
Velocity in $y$-direction is, $\frac{dy}{dt}=v_y=2 \mathrm{m} {\mathrm{s}}^{-1}$
Acceleration in $x$-direction is, $\frac{d^{2}x}{d{t}^2}=\frac{d{v}_x}{dt}=a_x=0$
Acceleration in $y$-direction is, $\frac{d^{2}y}{d{t}^2}=\frac{d{v}_y}{dt}=a_y=a$
Trajectory relation, $y=4x^2$
On differentiating trajectory relation, we have
$\Rightarrow \frac{dy}{dt}=4\times 2x\times \frac{dx}{dt}$
$\Rightarrow v_y=8x{v}_x$
Again differentiating, we have
$\Rightarrow \frac{d{v}_y}{dt}=8\left\{x\frac{d{v}_x}{dt}+v_x\frac{dx}{dt}\right\}$
$\Rightarrow a=8\left\{x\times a_x+v_x\times v_x\right\}$
$\Rightarrow a=8\left\{x\times 0+2\times 2\right\}$
$\Rightarrow a=8\times 4=32 \mathrm{m} {\mathrm{s}}^{-2}$