Question

In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision. because STATEMENT-2 In an elastic collision, the linear momentum of the system is conserved. (A) Statement- 1 is True, Statement- 2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement- 1 is False, Statement- 2 is True

Short answer

Both statements are true; however, the conservation of momentum (Statement-2) is not a complete explanation for the relative speed being equal before and after the collision (Statement-1). The correct choice is (B).

Step-by-step solution

Understanding Elastic Collisions

An elastic collision is a situation in which two bodies collide and bounce off each other without undergoing any deformation or heat generation. In such collisions, both the kinetic energy and the linear momentum of the system are conserved.

Analyzing Statement-1

Statement-1 says that the relative speed of the bodies after the collision is equal to the relative speed before the collision. This is indeed a characteristic of elastic collisions, as not only the momentum but also the kinetic energy (which is dependent on speed) is conserved.

Analyzing Statement-2

Statement-2 says that the linear momentum of the system is conserved in an elastic collision. This is a fundamental principle in physics called the conservation of momentum, and it holds true for elastic collisions.

Checking If Statement-2 Is the Correct Explanation for Statement-1

Even though both statements are true, the conservation of linear momentum alone does not guarantee that the relative speeds before and after the collision will be equal. For the relative speeds to be equal, the kinetic energy must also be conserved. Thus, Statement-2 is not a complete explanation for Statement-1, because conservation of kinetic energy also plays a role.

Key concepts

Conservation of Momentum

The principle of conservation of momentum is a cornerstone of physics and particularly essential when analyzing collisions. According to this law, the total momentum of a closed system remains constant if no external forces are acting on it. In a collision between two bodies, such as billiard balls or vehicles, the sum of their momenta before impact must equal the sum of their momenta after impact.

Momentum is a vector quantity, meaning it has both magnitude and direction, and is defined as the product of an object's mass and its velocity (\(p=mv\)). Consider two colliding objects in an isolated system: if one object transfers momentum to another during a collision, the first object loses as much momentum as the second object gains. This transfer results in momentum conservation throughout the process, a principle that holds even if the objects stick together or rebound apart in elastic and inelastic collisions alike.

In the context of the problem given, the conservation of momentum allows us to predict the post-collision velocities of the bodies involved, provided we know their masses and pre-collision velocities. However, this only explains the how of the speeds after collision, not the why—specifically, it doesn't alone explain why the relative speeds remain equal which involves kinetic energy conservation.

Kinetic Energy Conservation

Kinetic energy conservation in physics is closely related to elastic collisions. Kinetic energy is the energy an object possesses due to its motion and is expressed by the equation \( KE = \frac{1}{2}mv^2 \), where 'm' is mass and 'v' is velocity. During an elastic collision, not only is momentum conserved, but kinetic energy is also conserved. This means that the total kinetic energy of the system before collision is equal to the total kinetic energy after the collision.

This conservation results from the absence of energy conversion into other forms such as heat or sound, which might occur during an inelastic collision. For two bodies to have equal relative speeds before and after collision, as mentioned in Statement-1, it is the conservation of kinetic energy that provides a clear explanation.

In our problem, the kinetic energy's conservation implies a symmetry in the velocity changes of the colliding bodies, resulting in the same relative speed post-collision as pre-collision, aligning with the characteristic property of elastic collisions. Without this conservation, we could have scenarios where momentum is conserved but the kinetic energies are not, leading to different outcomes.

Physics of Collisions

Collisions in physics are broadly categorized into elastic and inelastic collisions, characterized by different energy conservation properties. Elastic collisions, the type under our focus, occur when objects collide and rebound without permanent deformation or heat generation, conserving both kinetic energy and momentum.

The physics of collisions involves analyzing the transfer of momentum and energy between the bodies involved. During an elastic collision, not only are the velocities of the objects altered, but this change happens in such a way that the relative speeds are unchanged, an attribute explained by the simultaneous conservation of kinetic energy and momentum.

In an inelastic collision, although momentum is still conserved, kinetic energy is not. Some of the kinetic energy is transformed into other forms of energy, such as heat or internal energy, which can result in the deformation of the colliding objects.

Understanding the distinction between elastic and inelastic collisions is paramount in predicting the final states of the objects after collision, which is integral to solving many real-world physics problems.

JEE Advanced Physics

The Joint Entrance Examination (JEE) Advanced is a highly competitive entrance exam for engineering colleges in India, and physics is a significant component of the test. The questions on the exam often include topics such as conservation of momentum and kinetic energy, as these principles govern a wide range of physical phenomena. A deep comprehension of these concepts, especially in the context of the physics of collisions, is vital for prospects aspiring to score well.

For instance, the problem at hand, dealing with the characteristics of an elastic collision, is a typical example of a JEE Advanced physics problem. It not only tests the student's knowledge of fundamental principles but also their ability to interconnect these principles to comprehend the complete picture of a physical situation.

To excel in JEE Advanced physics, a candidate must have a thorough understanding of not just the definitions, but also the application of conservation laws, and the ability to analyze different types of collisions and their consequences. A strong foundation in these areas empowers students to confidently tackle complex and challenging physics problems and achieve a high score.