The position of an object having mass \(0.1\) kg as a function of time \(t\) is given as \(\vec{r} = \left(10t^2\hat{i} + 5t^3\hat{j}\right)\) m. At \(t = 1\) s, which of the following statements are correct?
A. The linear momentum \(\vec{p} = \left(2\hat{i} + 1.5\hat{j}\right)\) kg·m/s.
B. The force acting on the object \(\vec{F} = \left(2\hat{i} + 3\hat{j}\right)\) N.
C. The angular momentum of the object about its origin \(\vec{L} = 15\hat{k}\) J·s.
D. The torque acting on the object about its origin \(\vec{\tau} = 20\hat{k}\) N·m.
Choose the correct answer from the options given below :

Given the position vector \(\vec{r} = 10t^2\hat{i} + 5t^3\hat{j}\)

Velocity \(\vec{v} = \dfrac{d\vec{r}}{dt} = 20t\hat{i} + 15t^2\hat{j}\)

Acceleration \(\vec{a} = \dfrac{d\vec{v}}{dt} = 20\hat{i} + 30t\hat{j}\)

At \(t = 1\) s:
\(\vec{r} = 10\hat{i} + 5\hat{j}\) m
\(\vec{v} = 20\hat{i} + 15\hat{j}\) m/s
\(\vec{a} = 20\hat{i} + 30\hat{j}\) m/s\(^2\)

Linear momentum \(\vec{p} = m\vec{v} = 0.1(20\hat{i} + 15\hat{j}) = 2\hat{i} + 1.5\hat{j}\) kg\(\cdot\)m/s (Statement A is correct)

Force \(\vec{F} = m\vec{a} = 0.1(20\hat{i} + 30\hat{j}) = 2\hat{i} + 3\hat{j}\) N (Statement B is correct)

Angular momentum \(\vec{L} = \vec{r} \times \vec{p} = (10\hat{i} + 5\hat{j}) \times (2\hat{i} + 1.5\hat{j}) = (10 \times 1.5 - 5 \times 2)\hat{k} = 5\hat{k}\) J\(\cdot\)s (Statement C is incorrect)

Torque \(\vec{\tau} = \vec{r} \times \vec{F} = (10\hat{i} + 5\hat{j}) \times (2\hat{i} + 3\hat{j}) = (10 \times 3 - 5 \times 2)\hat{k} = 20\hat{k}\) N\(\cdot\)m (Statement D is correct)

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

Answer: A, B and D only