91Ó°ÊÓ

In a perfectly elastic collision, the conservation of linear momentum is an essential concept that defines how objects interact during impact. It states that the total linear momentum, which is the product of mass and velocity, remains constant before and after the collision.

For two colliding objects, like disk A and disk B, we can set up an equation based on this principle:

• The momentum before collision: \(m_A \times (v_A)_1 + m_B \times 0\)
• The momentum after collision: \(m_A \times (v_A)_2 + m_B \times v_B\)

These equations show that the momentum lost by disk A is exactly transferred to disk B. By solving these equations, we can predict the velocities of both disks after the collision. This concept ensures that when the forces during collision are equal and opposite, momentum is conserved in the system.

The coefficient of restitution (usually denoted as 'e') measures how much kinetic energy remains for two bodies after a collision compared to before. This is particularly vital in elastic collisions where 'e' is equal to 1.

For disk A and disk B, the relationship provided by the coefficient of restitution is given by:

• \(e = 1 = \frac{v_B - (v_A)_2}{(v_A)_1 - 0}\)

This formula calculates the relative speed of separation over the relative speed of approach. In a perfectly elastic collision, an 'e' value of 1 signifies that the relative speed of separation post-collision remains the same as the relative speed at which they approached each other. This implies that no kinetic energy is lost during the collision, which characterizes a fully elastic impact.

For an elastic collision, one of the vital characteristics is the conservation of kinetic energy. This concept ensures that the total kinetic energy before the collision is the same as the total kinetic energy after the collision.

Let's examine the situation using the disks A and B:

• Before the collision: \(KE_1 = 0.5 \times m_A \times (v_A)_1^2 + 0.5 \times m_B \times 0^2\)
• After the collision: \(KE_2 = 0.5 \times m_A \times (v_A)_2^2 + 0.5 \times m_B \times v_B^2\)

In a perfectly elastic collision, these expressions are equal, representing the fact that kinetic energy is not lost but rather transferred between the objects. This property allows us to verify the accuracy of our calculated velocities. If the total kinetic energy remains unchanged, the solution is consistent with the physical situation described in the problem, confirming that an elastic collision has occurred.