Question

A body of mass 1kg begins to move under the action of a time dependent force \(\overset{⃗}{F} = (t\hat{i}+3t^2\hat{j}) \text{N}\) where \(\hat{i}\) and \(\hat{j}\) are the unit vectors along x and y axis. The power developed by the above force at time \(t = 2 \text{s}\) will be (in watt)

• A. \(100\)
• B. \(50\)
• C. \(25\)
• D. \(5\)

Hint:

Find velocity by integrating acceleration, then use P = F dot v.

Answer 1

The Correct Option is A

Step 1: Acceleration
With \(m = 1\ \text{kg}\), \(\vec a = \vec F = t\hat i + 3t^2\hat j\).

Step 2: Velocity
Starting from rest, \(\vec v = \frac{t^2}{2}\hat i + t^3\hat j\). At \(t=2\): \(\vec v = 2\hat i + 8\hat j\).

Step 3: Force at t = 2
\(\vec F = 2\hat i + 12\hat j\).

Step 4: Power
\[ P = \vec F\cdot\vec v = 2\times2 + 12\times8 = 4 + 96 = 100\ \text{W} \]
Option (A).

Final Answer:
The power is 100 W. \[ \boxed{\text{(A)}\ 100\ \text{W}} \]