1. Given data

• Mass of the object: $1000\,\text{g} = 1\,\text{kg}$
• Force: $\vec F = (2t\,\hat i + 3t^2\,\hat j)\,\text{N}$

We need power at time $t$.

2. Use Newton's second law to find acceleration

Since $\vec F = m\vec a$ and $m=1\,\text{kg}$, $\vec a = \vec F = 2t\,\hat i + 3t^2\,\hat j$

So, $a_x = 2t, \qquad a_y = 3t^2$

3. Find velocity by integrating acceleration

Assuming the object starts from rest at $t=0$,

For $x$-component: $v_x = \int 2t\,dt = t^2$

For $y$-component: $v_y = \int 3t^2\,dt = t^3$

Hence, $\vec v = t^2\,\hat i + t^3\,\hat j$

4. Use power formula

Instantaneous power is $P = \vec F \cdot \vec v$

So, $P = (2t\,\hat i + 3t^2\,\hat j) \cdot (t^2\,\hat i + t^3\,\hat j)$

$P = 2t \cdot t^2 + 3t^2 \cdot t^3$

$P = 2t^3 + 3t^5$

Thus, $\boxed{P = (2t^3 + 3t^5)\,\text{W}}$

5. Compare with options

• A: $(2t^2+18t^3)$ ❌
• B: $(3t^3+5t^5)$ ❌
• C: $(2t^3+3t^5)$ ✅
• D: $(2t^2+3t^3)$ ❌

Therefore, the correct option is C.

6. Comparison with stored answer

Stored correct answer: C

Our derived answer also gives C, so they agree.