Particle Force Problem
Question: At any instant the velocity of a particle of mass 1 kg is given by \[ \vec{v}(t) = (2t^2 \,\hat{i} + 3t \,\hat{j}) \quad \text{m/s}. \] If the force acting on the particle at \(t = 2\,\text{s}\) is \[ \vec{F}(2) = (x \,\hat{i} + 3 \,\hat{j}) \quad \text{N}, \] then find the value of \(x\).
- (1) 3
- (2) 4
- (3) 6
- (4) 8
Select the Correct Option
Detailed Step‑by‑Step Explanation
Step 1: Find the Acceleration
The velocity of the particle is given by:
\[
\vec{v}(t) = (2t^2 \,\hat{i} + 3t \,\hat{j}) \quad \text{m/s}.
\]
The acceleration is the time derivative of velocity:
\[
\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d}{dt}(2t^2) \,\hat{i} + \frac{d}{dt}(3t) \,\hat{j}.
\]
Calculating the derivatives:
\[
\frac{d}{dt}(2t^2) = 4t, \quad \frac{d}{dt}(3t) = 3.
\]
So,
\[
\vec{a}(t) = (4t\,\hat{i} + 3\,\hat{j}) \quad \text{m/s}^2.
\]
Step 2: Evaluate the Acceleration at \(t = 2\,\text{s}\)
Substitute \(t = 2\,\text{s}\) into the acceleration:
\[
\vec{a}(2) = (4 \times 2\,\hat{i} + 3\,\hat{j}) = (8\,\hat{i} + 3\,\hat{j}) \quad \text{m/s}^2.
\]
Step 3: Apply Newton's Second Law
Newton's second law states:
\[
\vec{F} = m \vec{a}.
\]
Given the mass \(m = 1\,\text{kg}\), the force at \(t = 2\,\text{s}\) is:
\[
\vec{F}(2) = 1 \times (8\,\hat{i} + 3\,\hat{j}) = (8\,\hat{i} + 3\,\hat{j}) \quad \text{N}.
\]
Step 4: Compare with the Given Force
The problem states that at \(t = 2\,\text{s}\), the force is:
\[
\vec{F}(2) = (x\,\hat{i} + 3\,\hat{j}) \quad \text{N}.
\]
Therefore, equating the components:
\[
x = 8.
\]
Final Answer: \(x = 8\), which corresponds to option (4).
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