91Ó°ÊÓ

In physics, the law of conservation of momentum states that the total momentum of a closed system remains constant over time, provided no external forces act upon it. In our exercise, we have two titanium spheres colliding elastically. This means that both momentum and kinetic energy are conserved during the collision.

Initially, each sphere has a momentum, given by the product of its mass and velocity. Because the spheres are moving toward each other with the same speed but in opposite directions, their individual momentums are:

• The first sphere: \( m_1(-v) \)
• The second sphere: \( m_2(v) \)

After the collision, one sphere comes to rest. This information helps us set up the conservation equation:

• Before collision: \(-m_1v + m_2v = m_2v_2^\prime \)
• After collision: \(m_1 \) is at rest, and \(v_2^\prime = v \)

By solving the momentum conservation equation, we find that both spheres must have equal mass, which is a crucial conclusion for understanding symmetric elastic collisions.

The center of mass of a system of particles is the point that moves as if all the system's mass were concentrated at that point and all external forces were applied there. For the two-sphere system in our original exercise, we consider their motion and momentum.

The center of mass velocity, \( v_{cm} \), is defined as the sum of the momentum of each sphere divided by the total system mass:\[ v_{cm} = \frac{m_1v_1 + m_2v_2}{m_1 + m_2} \]Initially, one sphere moves at \( -2 \text{ m/s} \) and the other at \( 2 \text{ m/s} \). The masses are equal, which simplifies the center of mass equation:

• Substituting values: \( v_{cm} = \frac{0.25(-2) + 0.25(2)}{0.25 + 0.25} = 0 \)

This result shows that the center of mass of the system remains stationary. It might seem surprising, but in an equal mass head-on elastic collision, this is a typical result. Understanding this concept helps us predict system behavior in collisions.

Kinetic energy is the energy possessed by an object due to its motion. In the case of elastic collisions, such as the one involving two titanium spheres from our problem, the total kinetic energy before and after the collision remains unchanged. This attribute distinguishes elastic collisions from inelastic ones.

Initially, each sphere has kinetic energy given by \( KE = \frac{1}{2}mv^2 \). So:

• The first sphere: \( KE_1 = \frac{1}{2} \times 0.25 \times (2^2) = 0.5 \text{ J} \)
• The second sphere: \( KE_2 = \frac{1}{2} \times 0.25 \times (2^2) = 0.5 \text{ J} \)

Overall, the system has a total kinetic energy of \( 1.0 \text{ J} \) initially.

After the collision, the sphere at rest has no kinetic energy, but the moving sphere's kinetic energy remains \( 0.5 \text{ J} \), ensuring that the system's total energy remains constant at \( 1.0 \text{ J} \). The conservation of kinetic energy confirms that the collision does not transform kinetic energy to any other form, like thermal or sound.