In an elastic collision of two billiard balls, which of the following quantities remain conserved during the s

NCERT Exemplar · Q28

Q. In an elastic collision of two billiard balls, which of the following quantities remain conserved during the short time of collision of the balls (i.e., when they are in contact).

(a) Kinetic energy.

(b) Total linear momentum? Give reason for your answer in each case.

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Concept understanding — Elastic Collision

Elastic Collision – From Intuition to Precision

Imagine two billiard balls on a table. They smash into each other, bounce apart, and keep moving. Now imagine two lumps of clay that collide and stick together — they stop moving as separate objects. The first case feels like nothing is lost; the second clearly loses motion. That feeling is the seed of the idea.

An elastic collision is the idealised version of that first case: a collision where no kinetic energy is turned into heat, sound, or permanent deformation. All the motion-energy that went in comes out again as motion-energy of the same total amount. The objects bounce perfectly.

The precise statement

For any collision between two bodies (call them $A$ and $B$ ), two things are always true if no external force acts:

Total linear momentum is conserved — this is a law of physics, always.
Total kinetic energy may or may not be conserved — that depends on the nature of the collision.

An elastic collision is defined by the second condition: both total momentum and total kinetic energy are conserved.

Let the masses be $m_1$ , $m_2$ and the velocities before collision be $u_1$ , $u_2$ ; after collision, $v_1$ , $v_2$ . Then:

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

These two equations together define an elastic collision.

What this means physically

In an elastic collision, the objects do not get dented, heated, or stuck. They exchange energy and momentum purely through reversible deformation — like two perfect springs that compress and then fully recover. Real collisions are never perfectly elastic (some energy always leaks into sound or heat), but many are close enough: billiard balls, steel ball bearings, gas molecules.

Watch out

A common mistake is to think "elastic" means the objects themselves are elastic (like a rubber band). That's not the point. The collision is elastic — the total kinetic energy is the same before and after. A rubber ball hitting a wall can be nearly elastic; a lump of clay hitting a wall is not.

A useful derived result

From the two conservation equations, you can derive a neat relation for one-dimensional collisions:

u_1 - u_2 = -(v_1 - v_2)

That is, the relative speed of approach equals the relative speed of separation. This is often easier to use than the full energy equation in problems. …

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During the short time of contact in an elastic collision, total linear momentum is conserved because the forces are internal, but kinetic energy is not conserved as it is temporarily converted into elastic potential energy due to deformation.

An elastic collision is defined by the conservation of both total kinetic energy and total linear momentum before and after the collision. However, the question specifically asks about the quantities conserved during the short time of collision when the balls are in contact. This distinction is crucial for understanding the underlying physics.

Let's consider the system of the two billiard balls.

1. Kinetic Energy (during contact)

When two billiard balls collide, they exert forces on each other, causing them to momentarily deform. This deformation means that the balls are compressed or distorted from their original shape.

Energy Transformation: During this deformation phase, some of the initial kinetic energy of the system is temporarily converted into elastic potential energy stored within the deformed balls.
Minimum Kinetic Energy: At the instant of maximum deformation, the relative velocity between the centers of mass of the balls becomes zero, and the kinetic energy of the system is at its minimum value (though not necessarily zero, as the center of mass of the system might still be moving).
Restoration: As the balls regain their original shape, this stored elastic potential energy is converted back into kinetic energy.
Conclusion: Since kinetic energy is continuously being converted into potential energy and then back again during the contact phase, the total kinetic energy of the system is not constant during this short interval. It is only conserved before the collision starts and after the collision ends.

Watch out

A common misconception is to assume kinetic energy is conserved throughout an elastic collision. It is conserved overall (initial K.E. = final K.E.), but not instantaneously during the deformation phase.

2. Total Linear Momentum (during contact)

Consider the system consisting of the two billiard balls. The forces that the balls exert on each other during the collision are internal forces to this system.

Internal Forces: According to Newton's third law, the force exerted by ball A on ball B is equal in magnitude and opposite in direction to the force exerted by ball B on ball A. These are action-reaction pairs. …

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Q.In an elastic collision of two billiard balls, which of the following quantities remain conserved during the short time of collision of the balls (i.e., when they are in contact).

Concept understanding — Elastic Collision

Elastic Collision – From Intuition to Precision

The precise statement

What this means physically

A useful derived result

1. Kinetic Energy (during contact)

2. Total Linear Momentum (during contact)

Concept: Conservation Laws During the Instant of Contact in a Collision

Q.In an elastic collision of two billiard balls, which of the following quantities remain conserved during the short time of collision of the balls (i.e., when they are in contact).

1. Kinetic Energy (during contact)

2. Total Linear Momentum (during contact)

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