91Ó°ÊÓ

Momentum is a fundamental concept in physics that describes the motion of bodies. In an elastic collision, the total momentum is conserved. This means that before and after the collision, the momentum of the entire system remains constant.

For our scenario of three spheres, we apply the conservation of momentum in two instances. The first occurs when sphere 1 collides with sphere 2, and the second occurs when sphere 2 collides with sphere 3.

During the first collision, sphere 1 initially moves with a velocity whereas sphere 2 is stationary. The equation for conservation of momentum becomes:

• Before collision: The total momentum = \( m_1 v_1 \) (since sphere 2 is at rest)
• After collision: The total momentum = \( m_1 u_1 + m_2 u_2 \)

Here, \( u_1 \) is the velocity of sphere 1 and \( u_2 \) is that of sphere 2 post-collision.

This principle similarly applies to the second collision between sphere 2 and sphere 3. The equations allow us to understand how velocities transfer between the spheres, crucial for determining the maximum velocity that sphere 3 can achieve.

In elastic collisions, not only is momentum conserved, but kinetic energy is conserved as well. Kinetic energy is the energy an object possesses due to its motion and is given by \( \frac{1}{2} mv^2 \). For perfectly elastic collisions, the total kinetic energy before and after the collision remains the same.

For the first collision between spheres 1 and 2, the kinetic energy equation is written as:

• Before collision: Total kinetic energy = \( \frac{1}{2} m_1 v_1^2 \)
• After collision: Total kinetic energy = \( \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 \)

Solving these equations for each collision helps us determine the post-collision velocities of the spheres.

Similarly, for the second collision between spheres 2 and 3, the conservation of kinetic energy allows us to find the velocities \( v_2 \) and \( v_3 \) after the impact. This ensures no kinetic energy is lost, which is typical of perfectly elastic interactions.

To maximize sphere 3's velocity, we need to consider the optimal mass of sphere 2. When sphere 2 gains velocity from sphere 1 and transfers it to sphere 3, the efficiency of this transfer is related to the masses of the objects involved.

Through physics principles, particularly the formulas derived from conservation laws, it is shown that the optimal mass of sphere 2 is crucial. For sphere 3 to achieve the maximum velocity, the mass \( m_2 \) of sphere 2 is computed using the geometric mean of the masses of spheres 1 and 3:

• Optimal mass of sphere 2: \( m_2 = \sqrt{m_1 m_3} \)

With given values, \( m_1 = 9 \text{ kg} \) and \( m_3 = 1 \text{ kg} \), it is calculated that \( m_2 = 3 \text{ kg} \). This mass ensures that sphere 3 obtains the highest possible velocity following the interactions.

This insight highlights how intricately connected the masses and velocities are in a system of elastic collisions, ensuring the most effective transfer of energy and momentum through optimal conditions.