Question

If the collision occurs at time \(t = 0\), the value of \(v_{\text{cm}} / (a\omega)\) will be .

Hint:

For elastic collisions, use momentum conservation and velocity exchange principles.

Answer 1

At \(t = 0\), the velocities of the particles are: \[ v_1 = a \omega \cos 0 = a \omega, \quad v_2 = -a \omega. \] After collision, velocity exchange occurs. The center of mass velocity is: \[ v_{\text{cm}} = \frac{m \cdot \frac{a\omega}{2} + m \cdot a\omega}{2m} = \frac{3a\omega}{4}. \] Thus: \[ \frac{v_{\text{cm}}}{a\omega} = 0.75. \]