At \(t=0\), a body of mass \(100\) g starts moving under the influence of a force \((5\hat{i}+10\hat{j})\) N. After \(2\) s its position is \((2x\hat{i}+5y\hat{j})\) m. The ratio \(x:y\) is ______.

Mass \(m = 100 \text{ g} = 0.1 \text{ kg}\)

Force \(\vec{F} = 5\hat{i} + 10\hat{j} \text{ N}\)

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

Since the body starts from rest, initial velocity \(\vec{u} = 0\).

Position after \(t = 2 \text{ s}\) is given by \(\vec{r} = \vec{u}t + \dfrac{1}{2}\vec{a}t^2\)

\(\vec{r} = \dfrac{1}{2}(50\hat{i} + 100\hat{j})(2)^2 = 100\hat{i} + 200\hat{j} \text{ m}\)

Comparing with the given position \(\vec{r} = 2x\hat{i} + 5y\hat{j}\):

\(2x = 100 \Rightarrow x = 50\)

\(5y = 200 \Rightarrow y = 40\)

The ratio \(x:y = 50:40 = 5:4\)

Answer: \(5:4\)