Question

A force \( \vec{F} = (5\hat{i} + 2\hat{j}) \, \mathrm{N} \) is acting for 2 s on an object of mass 0.1 kg, which is initially at rest at the origin. Find the final position.

• A. \( 50\hat{i} + 20\hat{j} \)
• B. \( 100\hat{i} + 20\hat{j} \)
• C. \( 50\hat{i} + 40\hat{j} \)
• D. \( 100\hat{i} + 40\hat{j} \)

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
According to Newton's Second Law, a constant force produces a constant acceleration. We can then use the equations of motion for constant acceleration to find the displacement, which, starting from the origin, gives the final position.
Step 2: Key Formula or Approach:
1. Acceleration \( \vec{a} = \frac{\vec{F}}{m} \).
2. Displacement \( \vec{s} = \vec{u}t + \frac{1}{2}\vec{a}t^2 \).
Since \( \vec{u} = 0 \), \( \vec{s} = \frac{1}{2}\vec{a}t^2 \).
Step 3: Detailed Explanation:
Given: \( \vec{F} = 5\hat{i} + 2\hat{j} \), \( m = 0.1 \, \text{kg} \), \( t = 2 \, \text{s} \). Calculate acceleration: \[ \vec{a} = \frac{5\hat{i} + 2\hat{j}}{0.1} = (50\hat{i} + 20\hat{j}) \, \text{m/s}^2 \] Calculate final position (displacement from origin): \[ \vec{r} = \frac{1}{2} (50\hat{i} + 20\hat{j})(2)^2 \] \[ \vec{r} = \frac{1}{2} (50\hat{i} + 20\hat{j})(4) \] \[ \vec{r} = 2(50\hat{i} + 20\hat{j}) = 100\hat{i} + 40\hat{j} \]
Step 4: Final Answer:
The final position of the object is \( 100\hat{i} + 40\hat{j} \).