A smooth sphere A is moving on a frictionless horizontal...

A smooth sphere A is moving on a frictionless horizontal plane with angular speed $\omega$ and centre of mass velocity $\mathrm{v}$. It collides elastically and head on with an identical sphere $\mathrm{B}$ at rest. Neglect friction everywhere. After the collision, their angular speeds are $\omega_{\mathrm{A}}$ and $\omega_{\mathrm{B}}$ respectively, Then $\{\mathrm{A}\} \omega_{\mathrm{A}}<\omega_{\mathrm{B}}$ $\{B\} \omega_{A}=\omega_{B} \quad\{C\} \omega_{A}=\omega \quad\{D\} \omega=\omega_{B}$

A smooth sphere A is moving on a frictionless horizontal plane with angular speed $\omega$ and centre of mass velocity $\mathrm{v}$. It collides elastically and head on with an identical sphere $\mathrm{B}$ at rest. Neglect friction everywhere. After the collision, their angular speeds are $\omega_{\mathrm{A}}$ and $\omega_{\mathrm{B}}$ respectively, Then
$\{\mathrm{A}\} \omega_{\mathrm{A}}<\omega_{\mathrm{B}}$
$\{B\} \omega_{A}=\omega_{B} \quad\{C\} \omega_{A}=\omega \quad\{D\} \omega=\omega_{B}$

Key Concepts

Elastic Collisions

An elastic collision is one in which not only is the total momentum of the system conserved, but the total kinetic energy is conserved as well. This principle is crucial when analyzing colliding bodies because it allows one to set up equations that balance both energy and momentum before and after the collision.

Conservation of Linear Momentum

In any collision free from external forces, the total linear momentum is conserved. This conservation law is pivotal in solving collision problems, as it relates the pre? and post-collision velocities of the objects involved and helps determine the outcome of the interaction.

Frictionless Surfaces

A frictionless surface implies that there is no resistance to motion along the plane and, more importantly, that no frictional forces act during contact. This absence of friction means that tangential impulses, which could alter rotational speeds (spin), are not present, thereby leaving the intrinsic rotation of bodies unaffected by the collision.

Decoupling of Rotational and Translational Motion

On a frictionless surface, the forces responsible for changing the translational motion of an object (like the collision impulse acting through the center of mass) do not create any torque since they do not act off-center. Consequently, the rotational motion (spin) of the object remains unchanged through the collision, effectively decoupling it from the translational dynamics.

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