Analysis of Kinematic and Dynamic Vectors of a Particle

Given mass $m = 0.1\text{ kg}$ and position vector $\vec{r}(t) = 10t^2\hat{i} + 5t^3\hat{j}\text{ m}$.

Differentiating $\vec{r}(t)$ with respect to time yields velocity $\vec{v}(t) = 20t\hat{i} + 15t^2\hat{j}\text{ m/s}$ and acceleration $\vec{a}(t) = 20\hat{i} + 30t\hat{j}\text{ m/s}^2$. At $t = 1\text{ s}$, the values are $\vec{r} = 10\hat{i} + 5\hat{j}$, $\vec{v} = 20\hat{i} + 15\hat{j}$, and $\vec{a} = 20\hat{i} + 30\hat{j}$.

1. Linear momentum: $\vec{p} = m\vec{v} = 0.1\left(20\hat{i} + 15\hat{j}\right) = 2\hat{i} + 1.5\hat{j}\text{ kg}\cdot\text{m/s}$ (Statement A is correct).
2. Force: $\vec{F} = m\vec{a} = 0.1\left(20\hat{i} + 30\hat{j}\right) = 2\hat{i} + 3\hat{j}\text{ N}$ (Statement B is correct).
3. Angular momentum: $\vec{L} = \vec{r} \times \vec{p} = (10\hat{i} + 5\hat{j}) \times (2\hat{i} + 1.5\hat{j}) = (15 - 10)\hat{k} = 5\hat{k}\text{ J s}$ (Statement C is incorrect).
4. Torque: $\vec{\tau} = \vec{r} \times \vec{F} = (10\hat{i} + 5\hat{j}) \times (2\hat{i} + 3\hat{j}) = (30 - 10)\hat{k} = 20\hat{k}\text{ N m}$ (Statement D is correct).

Therefore, statements A, B, and D only are correct.

Correct Option: D