Answer carefully, with reasons: (a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)? (b) Is the total linear momentum conserved during the short time of an elastic collision of two balls? (c) What are the answers to (a) and (b) for an inelastic collision? (d) If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential energy).

Answer carefully, with reasons: (a) In an elastic collision of two billiard balls, is the total kinetic energy

Exercises · 5.8

Q. Answer carefully, with reasons:

(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)?

(b) Is the total linear momentum conserved during the short time of an elastic collision of two balls?

(a) and

(b) for an inelastic collision?

(d) If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential energy).

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Concept understanding — Conservative Force Work

What is a Conservative Force? — Starting from Intuition

Imagine you are carrying a bucket of water up a hill and then walking back down. The work your muscles do against gravity depends only on how high you climbed, not on the path you took. Whether you go straight up the steep side or take a long winding road, the net work done by gravity on the bucket when you return to the starting point is exactly zero.

That is the core idea: a force is conservative if the work it does on an object moving between two points depends only on those points, not on the path taken.

The Precise Statement

\vec{F}

Mathematically:

W_{A \to B} = \int_{A}^{B} \vec{F} \cdot d\vec{r} \quad \text{is path-independent}

This single property leads to two equivalent, powerful consequences:

Work around any closed loop is zero. If you go from $A$ to $B$ along one path and return along another, the total work is zero:

\oint \vec{F} \cdot d\vec{r} = 0

The force can be written as the negative gradient of a potential energy function $U$ :

\vec{F} = -\nabla U

This means you can define a potential energy for the system — a stored energy that depends only on position.

Examples You Already Know

Force	Conservative?	Why
Gravity (near Earth)	Yes	Work depends only on height difference
Spring force ( $F = -kx$ )	Yes	Work depends only on stretch/compression
Electrostatic force	Yes	Work depends only on charge positions
Friction	No	Work depends on path length — longer path = more work
Air resistance	No	Same reason — dissipative

Watch out

A common mistake: thinking "conservative" means the force conserves kinetic energy. It does not. It means the force itself allows a potential energy to be defined, so total mechanical energy (kinetic + potential) is conserved when only conservative forces act.

Why This Matters for Exams

When you see a problem with gravity, springs, or electric fields, you can immediately:

Use energy conservation: $K_i + U_i = K_f + U_f$
Ignore the path — only initial and final positions matter
Compute work as $W = -\Delta U$ instead of doing a line integral …

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During the short contact time of any collision (elastic or inelastic), kinetic energy is generally not conserved as it converts to potential energy of deformation or heat, but total linear momentum is conserved if no net external force acts on the system. If potential energy depends only on separation, the collision is elastic.

Let's break down the concepts of elastic and inelastic collisions, focusing on what happens during the brief moment the objects are actually in contact. This distinction is crucial for understanding conservation laws.

Understanding Collisions: Before, During, and After

A collision is an event where two or more bodies exert forces on each other for a relatively short time. We typically analyze collisions by comparing the state before the collision to the state after the collision. However, the question specifically asks about the state during the short time of contact.

The key difference between elastic and inelastic collisions lies in how energy is handled during this contact phase:

Elastic Collision: Kinetic energy is temporarily converted into potential energy of deformation (like compressing a spring) and then fully converted back into kinetic energy. No energy is lost as heat, sound, or permanent deformation.
Inelastic Collision: Kinetic energy is converted into potential energy of deformation, but not all of it is recovered as kinetic energy. Some energy is permanently converted into other forms like heat, sound, or permanent deformation of the objects.

Now, let's address each part of the question.

(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)?

Definition of Elastic Collision: An elastic collision is defined by the conservation of total kinetic energy before and after the collision. This means the system's total kinetic energy remains the same when comparing the initial state (just before contact) and the final state (just after separation).
What happens during contact? When two billiard balls collide, they momentarily deform. Imagine two soft balls hitting each other – they flatten slightly at the point of contact. This deformation requires work to be done against the internal elastic forces of the balls.
Energy Transformation: During this deformation phase, the kinetic energy of the colliding balls is temporarily converted into elastic potential energy stored within the deformed material of the balls. At the point of maximum compression, the relative velocity of the centers of mass of the balls becomes zero, and a significant portion of the initial kinetic energy is stored as potential energy. As the balls then push apart, this stored potential energy is converted back into kinetic energy.
Conclusion for Kinetic Energy during contact: Since kinetic energy is being converted into potential energy (and then back again), the total kinetic energy of the system is not constant during the actual contact time. It decreases as potential energy increases, reaches a minimum at maximum compression, and then increases again as potential energy decreases.

Watch out

Do not confuse "total kinetic energy is conserved in an elastic collision" (referring to before vs. after) with "total kinetic energy is conserved at every instant during the collision." The latter is generally false.

Therefore, the total kinetic energy is not conserved during the short time of contact in an elastic collision.

(b) Is the total linear momentum conserved during the short time of an elastic collision of two balls?

Condition for Momentum Conservation: The total linear momentum of a system is conserved if and only if the net external force acting on the system is zero.
Forces during Collision: When two billiard balls collide, the forces they exert on each other (the action-reaction pair) are internal forces to the system consisting of the two balls.
External Forces: Assuming the collision occurs on a frictionless surface (or we consider the vertical forces of gravity and normal force to cancel out), there are no significant external horizontal forces acting on the system of the two balls during the collision.
Conclusion for Linear Momentum during contact: Since the forces involved in the collision are internal to the system, and assuming no net external force acts on the system, the total linear momentum of the system of the two balls is conserved throughout the entire collision process, including the short time they are in contact.

Therefore, the total linear momentum is conserved during the short time of contact in an elastic collision.

(c) What are the answers to (a) and (b) for an inelastic collision?

Kinetic Energy in Inelastic Collision (during contact):
- Similar to an elastic collision, during the contact phase of an inelastic collision, the balls deform. Kinetic energy is converted into potential energy of deformation.
- However, in an inelastic collision, some of this energy is irreversibly converted into other forms like heat, sound, or permanent deformation. This means that even at maximum compression, not all the initial kinetic energy is stored as recoverable potential energy, and certainly, not all of it will be converted back to kinetic energy after separation.
- Thus, the total kinetic energy of the system is definitely not conserved during the short time of contact. It decreases and does not fully recover.
Linear Momentum in Inelastic Collision (during contact):
- The principle of linear momentum conservation depends only on the absence of net external forces, not on the type of collision (elastic or inelastic).
- As explained in part (b), the forces between the colliding balls are internal forces. …

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Q.Answer carefully, with reasons:

Concept understanding — Conservative Force Work

What is a Conservative Force? — Starting from Intuition

The Precise Statement

Examples You Already Know

Why This Matters for Exams

Understanding Collisions: Before, During, and After

(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)?

(b) Is the total linear momentum conserved during the short time of an elastic collision of two balls?

(c) What are the answers to (a) and (b) for an inelastic collision?

Concept: Momentum vs. Kinetic Energy Conservation During Contact

Q.Answer carefully, with reasons:

Understanding Collisions: Before, During, and After

(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)?

(b) Is the total linear momentum conserved during the short time of an elastic collision of two balls?

(c) What are the answers to (a) and (b) for an inelastic collision?

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