Question

If force \( \vec{F}= 2t\,\hat{i} + 3t^2\,\hat{j} \) acts on a particle of mass \(m = 2\,\text{kg}\), find the power at \(t = 2\,\text{s}\) if the particle was initially at rest.

• A. \(54\,\text{W}\)
• B. \(58\,\text{W}\)
• C. \(56\,\text{W}\)
• D. \(52\,\text{W}\)

Answer 1

The Correct Option is C

Concept:
Power delivered by a force is given by the dot product of force and velocity. \[ P = \vec{F} \cdot \vec{v} \] Acceleration is related to force by Newton's second law: \[ \vec{a} = \frac{\vec{F}}{m} \] Velocity can be obtained by integrating acceleration. Step 1: Find acceleration. \[ \vec{a} = \frac{\vec{F}}{m} \] \[ \vec{a} = \frac{2t}{2}\hat{i} + \frac{3t^2}{2}\hat{j} \] \[ \vec{a} = t\,\hat{i} + \frac{3t^2}{2}\hat{j} \] Step 2: Find velocity by integrating acceleration. \[ \vec{v} = \int \vec{a}\,dt \] \[ \vec{v} = \int t\,dt\,\hat{i} + \int \frac{3t^2}{2}\,dt\,\hat{j} \] \[ \vec{v} = \frac{t^2}{2}\hat{i} + \frac{t^3}{2}\hat{j} \] Step 3: Compute power. \[ P = \vec{F} \cdot \vec{v} \] \[ P = (2t\,\hat{i} + 3t^2\,\hat{j}) \cdot \left(\frac{t^2}{2}\hat{i} + \frac{t^3}{2}\hat{j}\right) \] \[ P = t^3 + \frac{3}{2}t^5 \] Step 4: Substitute \(t = 2\,s\). \[ P = 2^3 + \frac{3}{2}(2^5) \] \[ P = 8 + \frac{3}{2}\times 32 \] \[ P = 8 + 48 \] \[ P = 56\,\text{W} \]