91Ó°ÊÓ

In physics, the principle of conservation of linear momentum states that if no external forces act on a system, the total momentum of the system remains constant. This is particularly evident in elastic collisions, where none of the kinetic energy or momentum is dissipated. Linear momentum is given by the product of mass and velocity, so in any collision, the sum of the momentum of all objects before the collision must equal the sum of the momentum of all objects after the collision.

In the exercise involving spheres A and B, sphere A initially has momentum since it is moving. When it collides with stationary sphere B, because no external force acts on the system and due to the nature of elastic collisions, sphere B picks up the entire velocity that sphere A had before impact. Consequently, sphere A stops moving, and sphere B moves forward at the same speed that A initially had. This showcases the conservation of linear momentum very clearly.

Remember:

• Momentum before collision = Momentum after collision
• For collisions of identical spheres, the moving sphere stops and the stationary sphere takes on its speed

Kinetic energy in the context of motion is the energy that an object possesses due to its motion, which is derived from its mass and velocity. In elastic collisions, the total kinetic energy of the system is conserved, meaning it remains the same before and after the collision.

Mathematically, kinetic energy (KE) is expressed as \[ KE = \frac{1}{2} mv^2 \] where \( m \) is the mass of an object and \( v \) is its velocity.

In our example with spheres A and B, all the kinetic energy of sphere A is transferred to sphere B during the collision. As a result, sphere B moves with the same velocity \( u \), and sphere A comes to a stop, thus conserving the total kinetic energy of the system before and after the impact. This illustrates perfectly how kinetic energy behaves during elastic collisions.

Key Points:

• Total kinetic energy is conserved in elastic collisions
• Spheres trade speeds to maintain constant kinetic energy in a frictionless system

Rotational motion refers to the motion of a body around an internal axis. Unlike linear motion, which is affected by velocities in a straight line, rotational motion involves angular velocity, represented by \( \omega \). When analyzing elastic collisions, particularly those without friction, the rotational motion can be completely independent of the linear motion.

In the problem of the colliding spheres, sphere A not only moves but also rotates. Because there is no friction between the surfaces of spheres A and B, the rotational motion of sphere A is unaffected by the linear collision. This means that even after sphere A comes to rest concerning its linear motion, it continues to rotate with its original angular velocity \( \omega \). Sphere B, meanwhile, only picks up linear speed because there is no mechanism to transfer rotational energy between the spheres in the absence of friction.

Where rotational motion is concerned:

• Without friction, rotation is unaffected in collisions
• Each sphere's rotation stays as it was before the impact