Ratio of Position Coordinates for Mass Under Constant Vector Force

To find the ratio $x : y$, we analyze the motion of the body under the action of a constant force.

1. Given Data:

• Mass of the body, $m = 100\text{ g} = 0.1\text{ kg}$
• Applied force, $\vec{F} = (5\hat{i} + 10\hat{j})\text{ N}$
• Time, $t = 2\text{ s}$
• Initial velocity, $\vec{u} = 0\text{ m/s}$ (starts moving from rest at $t=0$)
• Position vector at $t = 2\text{ s}$, $\vec{r} = (2x\hat{i} + 5y\hat{j})\text{ m}$

2. Calculation of Acceleration: Using Newton's second law of motion, $\vec{F} = m\vec{a}$: $\vec{a} = \frac{\vec{F}}{m} = \frac{5\hat{i} + 10\hat{j}}{0.1} = (50\hat{i} + 100\hat{j})\text{ m/s}^2$

3. Calculation of Position Vector at $t = 2\text{ s}$: Using the second equation of motion for constant acceleration: $\vec{r} = \vec{u}t + \frac{1}{2}\vec{a}t^2$

Since $\vec{u} = 0$: $\vec{r} = \frac{1}{2}(50\hat{i} + 100\hat{j})(2)^2$ $\vec{r} = 2(50\hat{i} + 100\hat{j}) = (100\hat{i} + 200\hat{j})\text{ m}$

4. Equating Coordinates: Comparing the calculated position vector with the given position vector $\vec{r} = 2x\hat{i} + 5y\hat{j}$:

Along the $x$-axis: $2x = 100 \implies x = 50$

Along the $y$-axis: $5y = 200 \implies y = 40$

5. Ratio $x : y$: $\frac{x}{y} = \frac{50}{40} = \frac{5}{4}$

Thus, the ratio $x : y$ is $5 : 4$.

Correct Option: D