1. Given data

• Mass of body: $m = 2\,\text{kg}$
• Initial velocity: $\vec{v}_{in} = 3\hat{i} + 4\hat{j}\;\text{m s}^{-1}$
• Force acting on body: $\vec{F} = 6\hat{k}\;\text{N}$
• Time spent in field: $t = \frac{5}{3}\;\text{s}$

2. Find acceleration

Using Newton's second law, $\vec{a} = \frac{\vec{F}}{m} = \frac{6\hat{k}}{2} = 3\hat{k}\;\text{m s}^{-2}$

So the acceleration is only along the positive $z$-axis.

3. Find change in velocity

Change in velocity is $\Delta \vec{v} = \vec{a}t = 3\hat{k} \times \frac{5}{3} = 5\hat{k}\;\text{m s}^{-1}$

4. Find final velocity

$\vec{v}_{out} = \vec{v}_{in} + \Delta \vec{v}$

$\vec{v}_{out} = (3\hat{i} + 4\hat{j}) + 5\hat{k}$

$\boxed{\vec{v}_{out} = 3\hat{i} + 4\hat{j} + 5\hat{k}\;\text{m s}^{-1}}$

5. Check options

• A: $3\hat{i} + 4\hat{j} + \sqrt{5}\hat{k}$ ❌
• B: $4\hat{i} + 3\hat{j} + 5\hat{k}$ ❌
• C: $3\hat{i} + 4\hat{j} - 5\hat{k}$ ❌
• D: $3\hat{i} + 4\hat{j} + 5\hat{k}$ ✅

Therefore, the correct option is: $\boxed{\text{D}}$