Question

An object of mass \( 3 \, \text{kg} \) is at rest. Now a force of \( \vec{F} = 6t^2 \hat{i} + 4t \hat{j} \) is applied on the object. The velocity of the object at \( t = 3 \, \text{s} \) is:

• A. \( 18 \hat{i} + 3 \hat{j} \)
• B. \( 18 \hat{i} + 6 \hat{j} \)
• C. \( 3 \hat{i} + 18 \hat{j} \)
• D. \( 18 \hat{i} + 4 \hat{j} \)

Hint:

Key points to remember:
- Acceleration is the time derivative of velocity
- Velocity is obtained by integrating acceleration
- Initial conditions are crucial for solving differential equations
- For time-dependent forces, integration is necessary

Answer 1

The Correct Option is B

Step 1: Understand the given information
We have: - Mass \( m = 3 \, \text{kg} \) - Initial velocity \( \vec{v}(0) = \vec{0} \) (at rest) - Force \( \vec{F} = 6t^2 \hat{i} + 4t \hat{j} \, \text{N} \) - Time \( t = 3 \, \text{s} \)
Step 2: Apply Newton's second law
Using \( \vec{F} = m\vec{a} \), we get: \[ \vec{a} = \frac{\vec{F}}{m} = \frac{6t^2}{3} \hat{i} + \frac{4t}{3} \hat{j} = 2t^2 \hat{i} + \frac{4}{3}t \hat{j} \, \text{ms}^{-2} \]
Step 3: Find velocity by integrating acceleration
Velocity is the integral of acceleration: \[ \vec{v}(t) = \int \vec{a} \, dt = \left( \int 2t^2 \, dt \right) \hat{i} + \left( \int \frac{4}{3}t \, dt \right) \hat{j} \] \[ = \left( \frac{2}{3}t^3 + C_1 \right) \hat{i} + \left( \frac{2}{3}t^2 + C_2 \right) \hat{j} \]
Step 4: Apply initial condition
At \( t = 0 \), \( \vec{v} = \vec{0} \), so \( C_1 = C_2 = 0 \). Thus: \[ \vec{v}(t) = \frac{2}{3}t^3 \hat{i} + \frac{2}{3}t^2 \hat{j} \]
Step 5: Calculate velocity at \( t = 3 \, \text{s} \)
\[ \vec{v}(3) = \frac{2}{3}(3)^3 \hat{i} + \frac{2}{3}(3)^2 \hat{j} \] \[ = \frac{2}{3}(27) \hat{i} + \frac{2}{3}(9) \hat{j} \] \[ = 18 \hat{i} + 6 \hat{j} \, \text{ms}^{-1} \]