Question:
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______.
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______.
Updated On: Apr 10, 2026
- 1 : 2
- 2 : 5
- 5 : 2
- 5 : 4
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
By utilizing Newton's second law, we can determine the 2D acceleration vector of the mass. Since the force is constant and the body starts from rest, we can use the kinematic equations for uniform acceleration to find its position vector at $t = 2$s and compare it with the given coordinate expressions.
Step 2: Key Formula or Approach:
Newton's Second Law: $\vec{a} = \frac{\vec{F}}{m}$
Kinematics (from rest, $\vec{u} = 0$): $\vec{s} = \frac{1}{2}\vec{a}t^2$
Step 3: Detailed Explanation:
Mass $m = 100 \text{ g} = 0.1 \text{ kg}$.
Force $\vec{F} = 5\hat{i} + 10\hat{j} \text{ N}$.
Calculate the acceleration vector:
$\vec{a} = \frac{\vec{F}}{m} = \frac{5\hat{i} + 10\hat{j}}{0.1} = 50\hat{i} + 100\hat{j} \text{ m/s}^2$.
The body starts from rest, so initial velocity $\vec{u} = 0$.
The position vector after $t = 2$ s is:
$\vec{s} = \vec{u}t + \frac{1}{2}\vec{a}t^2 = 0 + \frac{1}{2} (50\hat{i} + 100\hat{j}) (2)^2$
$\vec{s} = \frac{1}{2} (50\hat{i} + 100\hat{j}) \times 4 = 2(50\hat{i} + 100\hat{j})$
$\vec{s} = 100\hat{i} + 200\hat{j} \text{ m}$.
We are given the position after 2 seconds as $(2x\hat{i} + 5y\hat{j})$ m.
Equating the components:
x-component: $2x = 100 \implies x = 50$.
y-component: $5y = 200 \implies y = 40$.
Find the ratio $x : y$:
$\frac{x}{y} = \frac{50}{40} = \frac{5}{4}$.
Step 4: Final Answer:
The ratio is 5 : 4.
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