A force on an object of mass $$100$$ g is $$(10\hat{i} + 5\hat{j})$$ N. The position of that object at $$t = 2$$ s is $$a\hat{i} + b\hat{j}$$ m after starting from rest. The value of $$\frac{a}{b}$$ will be ______.

Given: Mass $$m = 100$$ g $$= 0.1$$ kg, Force $$\vec{F} = (10\hat{i} + 5\hat{j})$$ N, initial velocity = 0 (starts from rest), $$t = 2$$ s.

Find the acceleration: Using Newton's second law $$\vec{F} = m\vec{a}$$:

$$\vec{a} = \frac{\vec{F}}{m} = \frac{10\hat{i} + 5\hat{j}}{0.1} = (100\hat{i} + 50\hat{j}) \text{ m/s}^2$$

Find the position at $$t = 2$$ s: Since the object starts from rest ($$\vec{u} = 0$$), using $$\vec{s} = \vec{u}t + \frac{1}{2}\vec{a}t^2$$:

$$\vec{s} = \frac{1}{2}(100\hat{i} + 50\hat{j})(2)^2$$

$$\vec{s} = \frac{1}{2}(100\hat{i} + 50\hat{j})(4)$$

$$\vec{s} = (200\hat{i} + 100\hat{j}) \text{ m}$$

So $$a = 200$$ and $$b = 100$$.

Find $$\frac{a}{b}$$: $$\frac{a}{b} = \frac{200}{100} = 2$$

The answer is 2.