Step-by-step Analysis

1. Analyze Statement 2: Statement 2 says: "In an elastic collision, the linear momentum of the system is conserved." The law of conservation of linear momentum states that if the net external force on a system of particles is zero, the total linear momentum of the system remains constant. A collision is a process that occurs over a very short time interval, where the forces the colliding bodies exert on each other (internal forces) are much larger than any external forces (like gravity or air resistance). Therefore, the net external force on the two-body system can be considered negligible. As a result, the total linear momentum of the system is conserved for any type of collision (elastic, inelastic, or perfectly inelastic) in the absence of significant external forces. Hence, Statement 2 is True.

2. Analyze Statement 1: Statement 1 says: "In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision." This property is commonly known as Newton's law of restitution for a perfectly elastic collision, where the coefficient of restitution $e=1$. It is mathematically expressed as $|\vec{v}_2 - \vec{v}_1| = |\vec{u}_1 - \vec{u}_2|$, where $\vec{u}_1, \vec{u}_2$ are initial velocities and $\vec{v}_1, \vec{v}_2$ are final velocities.

   Let's examine the conditions under which this is derived: The derivation requires both: (a) Conservation of linear momentum: $m_1 \vec{u}_1 + m_2 \vec{u}_2 = m_1 \vec{v}_1 + m_2 \vec{v}_2$ (b) Conservation of kinetic energy: $\frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2$

   The crucial point lies in the interpretation of "kinetic energy" and "bodies".

   • If the "bodies" are treated as point masses (or smooth spheres colliding centrally), then the only form of kinetic energy is translational. In this simplified model, an elastic collision conserves translational kinetic energy, and Statement 1 is indeed true.
   • However, the question uses the term "bodies", which can be extended rigid objects capable of rotation. The general definition of an elastic collision is one where the total kinetic energy is conserved. The total kinetic energy is the sum of translational and rotational kinetic energy: $K_{total} = K_{trans} + K_{rot}$.
   • In a collision between extended bodies (e.g., an off-center collision or a collision with friction), it is possible for some of the initial translational kinetic energy to be converted into rotational kinetic energy. For instance, two non-rotating bodies can start to rotate after colliding. $K_{i, trans} + K_{i, rot} = K_{f, trans} + K_{f, rot}$ If the bodies are not rotating initially ($K_{i, rot} = 0$) but acquire rotation ($K_{f, rot} > 0$), then to conserve the total energy, the final translational kinetic energy must be less than the initial translational kinetic energy ($K_{f, trans} < K_{i, trans}$).
   • The property that relative speed is conserved (Statement 1) is a direct consequence of the conservation of translational kinetic energy. Since translational kinetic energy is not necessarily conserved in a general elastic collision of extended bodies, Statement 1 is not universally true.

   Therefore, in the general case for "bodies", Statement 1 is False.

3. Conclusion:

   • Statement 1 is False (as it's not true for all elastic collisions involving extended bodies).
   • Statement 2 is True (linear momentum is conserved in any collision within an isolated system).

   This corresponds to option D.