1. Given position vector

   The position of the ant is $\vec S = 2t^2\,\hat j + 5\,\hat k$

   This means:

   • $y$-coordinate $= 2t^2$
   • $z$-coordinate $= 5$

   So the ant moves only along the $y$-direction, while $z$ remains constant.

2. Find velocity vector

   Velocity is the time derivative of position: $\vec v = \frac{d\vec S}{dt}$

   Differentiating, $\vec v = \frac{d}{dt}(2t^2\hat j + 5\hat k)$ $\vec v = 4t\hat j + 0\hat k$ $\vec v = 4t\hat j$

3. At $t=1\,$s

   Substitute $t=1$: $\vec v = 4(1)\hat j = 4\hat j\ \text{m/s}$

4. Magnitude and direction

   • Magnitude of velocity: $|\vec v| = 4\ \text{m/s}$
   • Direction: along $\hat j$, i.e. $y$-direction

5. Check options

   • A: $16\,\text{m/s}$ in $y$-direction $\rightarrow$ incorrect
   • B: $4\,\text{m/s}$ in $x$-direction $\rightarrow$ incorrect
   • C: $9\,\text{m/s}$ in $z$-direction $\rightarrow$ incorrect
   • D: $4\,\text{m/s}$ in $y$-direction $\rightarrow$ correct

Therefore, the correct answer is D.