Question

A particle of mass 1 kg, initially resting at origin, starts moving under the influence of a force \( \mathbf{F} = 4t^3 \hat{i} - 3t^2 \hat{j} \). If the speed of the particle at \( t = 1 \) is \( \sqrt{2} \), then the value of \( \alpha \) is

Hint:

To calculate the speed from a given force, integrate the force components to find velocity, and then calculate the magnitude of the velocity vector.

Answer 1

Correct Answer: 2

Step 1: Find the velocity components.
The velocity \( v \) is the integral of the force with respect to time: \[ \mathbf{v} = \int \mathbf{F} \, dt \] Substituting the force components \( \mathbf{F} = 4t^3 \hat{i} - 3t^2 \hat{j} \), we get: \[ v_x = \int 4t^3 \, dt = t^4 \] \[ v_y = \int -3t^2 \, dt = -t^3 \] Step 2: Find the speed of the particle.
The speed \( v \) is the magnitude of the velocity: \[ v = \sqrt{v_x^2 + v_y^2} \] At \( t = 1 \), we have: \[ v = \sqrt{(1)^4 + (-1)^3} = \sqrt{2} \] Step 3: Conclusion.
The value of \( \alpha \) is 2.