Question

At any instant the velocity of a particle of mass 500g is \( \left( 2t \hat{i} + 3t^2 \hat{j} \right) \, \text{ms}^{-1} \). If the force acting on the particle at \( t = 1 \) s is \( \left( \hat{i} + x \hat{j} \right) \, \text{N} \), then the value of \( x \) will be:

• A. 3
• B. 4
• C. 6
• D. 2

Hint:

To find the acceleration, take the derivative of the velocity with respect to time. The force is then found by multiplying the mass with the acceleration.

Answer 1

The Correct Option is A

Step 1: Using Newton's second law, \( \vec{F} = m \vec{a} \), where \( m \) is the mass and \( \vec{a} \) is the acceleration. The given velocity of the particle is: \[ \vec{v} = (2t \hat{i} + 3t^2 \hat{j}) \, {ms}^{-1} \] Step 2: The acceleration is the time derivative of the velocity: \[ \vec{a} = \frac{d\vec{v}}{dt} = \frac{d}{dt} \left( 2t \hat{i} + 3t^2 \hat{j} \right) \] Taking the derivative: \[ \vec{a} = 2 \hat{i} + 6t \hat{j} \] Step 3: Substituting \( t = 1 \) s into the acceleration expression: \[ \vec{a} = 2 \hat{i} + 6 \hat{j} \, {ms}^{-2} \] Step 4: The force \( \vec{F} \) is given as: \[ \vec{F} = m \vec{a} = 0.5 \, {kg} \times \left( 2 \hat{i} + 6 \hat{j} \right) = 1 \hat{i} + 3 \hat{j} \, {N} \] Step 5: Comparing this with the given force \( \vec{F} = \left( \hat{i} + x \hat{j} \right) \, {N} \), we can see that: \[ x = 3 \] Thus, the value of \( x \) is 3.