In An Elastic Collision What Is Conserved

An elastic collision is a fundamental concept in physics where two or more bodies collide without any net loss of kinetic energy in the system. Understanding what is conserved during such a collision is crucial for grasping various physical phenomena, from billiard balls colliding to the interactions of atoms and molecules.

Defining Elastic Collision

In the realm of physics, a collision is considered elastic if, and only if, it conserves both momentum and kinetic energy. This means that the total momentum and the total kinetic energy of the objects before the collision are equal to their respective totals after the collision. In simpler terms, no energy is lost to heat, sound, or deformation during the impact.

Key Characteristics

• Conservation of Kinetic Energy: This is the hallmark of an elastic collision. The sum of the kinetic energies of all colliding bodies remains constant.
• Conservation of Momentum: The total momentum of the system is conserved. Momentum, a vector quantity, is the product of an object's mass and velocity.
• No Energy Dissipation: Elastic collisions do not convert kinetic energy into other forms of energy like heat, sound, or potential energy.
• Idealization: In reality, perfectly elastic collisions are rare. Most real-world collisions involve some energy loss and are thus classified as inelastic.

What is Conserved?

Momentum

Momentum, denoted as p, is the product of an object's mass (m) and its velocity (v). Mathematically, p = mv. The law of conservation of momentum states that the total momentum of a closed system (one not affected by external forces) remains constant if no external forces act on the system.

Mathematical Representation

For a two-body collision, the conservation of momentum can be expressed as:

m1v1i + m2v2i = m1v1f + m2v2f

Where:

• m1 and m2 are the masses of the two bodies.
• v1i and v2i are the initial velocities of the two bodies.
• v1f and v2f are the final velocities of the two bodies.

This equation indicates that the total momentum before the collision (left side) is equal to the total momentum after the collision (right side).

Importance

The conservation of momentum is crucial in analyzing collisions because it provides a direct relationship between the masses and velocities of objects before and after the collision, regardless of the details of the collision process.

Kinetic Energy

Kinetic energy, denoted as KE, is the energy possessed by an object due to its motion. It is given by the formula KE = (1/2)mv^2, where m is the mass and v is the velocity of the object.

Mathematical Representation

In an elastic collision, the total kinetic energy before and after the collision remains the same. For a two-body collision, this can be expressed as:

(1/2)m1v1i^2 + (1/2)m2v2i^2 = (1/2)m1v1f^2 + (1/2)m2v2f^2

Where:

• m1 and m2 are the masses of the two bodies.
• v1i and v2i are the initial velocities of the two bodies.
• v1f and v2f are the final velocities of the two bodies.

This equation signifies that the sum of the kinetic energies of the two bodies before the collision equals the sum of their kinetic energies after the collision.

Significance

The conservation of kinetic energy, along with the conservation of momentum, provides a complete description of an elastic collision, allowing for the determination of the final velocities of the objects involved.

Contrasting with Inelastic Collisions

To fully appreciate the significance of conservation in elastic collisions, it is helpful to contrast them with inelastic collisions.

Inelastic Collisions

In an inelastic collision, kinetic energy is not conserved. Some of the kinetic energy is converted into other forms of energy, such as heat, sound, or deformation. While momentum is still conserved (assuming no external forces), the total kinetic energy after the collision is less than the total kinetic energy before the collision.

Types of Inelastic Collisions

• Perfectly Inelastic Collision: This is the extreme case where the colliding objects stick together after the collision, moving as one combined mass. A classic example is a bullet embedding itself in a block of wood.
• Partially Inelastic Collision: In this case, the objects do not stick together, but some kinetic energy is still lost. Examples include car crashes, where energy is dissipated through deformation and heat.

Key Differences

Real-World Examples

While perfectly elastic collisions are theoretical idealizations, several real-world scenarios closely approximate elastic collisions.

Billiard Balls

The collision of billiard balls is a classic example often cited to illustrate elastic collisions. When one billiard ball strikes another, very little kinetic energy is lost to friction or deformation. Most of the kinetic energy is transferred from one ball to the other, resulting in motion that appears to conserve both momentum and kinetic energy.

Atomic and Molecular Collisions

In the microscopic world, collisions between atoms and molecules can often be approximated as elastic, especially in ideal gases. The kinetic energy of these particles is conserved during collisions, which is crucial for understanding the behavior of gases and other thermodynamic systems.

Bouncing Balls

A bouncing ball provides a practical, albeit imperfect, example of an elastic collision. A perfectly elastic ball would bounce back to its original height without any loss of energy. However, real-world balls lose some energy due to air resistance and the ball's deformation upon impact, making the collision partially inelastic.

Mathematical Derivations and Problem-Solving

Understanding the mathematical relationships governing elastic collisions is essential for solving problems in physics.

Deriving Final Velocities

Consider two objects with masses m1 and m2, initial velocities v1i and v2i, and final velocities v1f and v2f. Using the conservation of momentum and kinetic energy, we can derive equations to solve for the final velocities.

From the conservation of momentum:

m1v1i + m2v2i = m1v1f + m2v2f

From the conservation of kinetic energy:

(1/2)m1v1i^2 + (1/2)m2v2i^2 = (1/2)m1v1f^2 + (1/2)m2v2f^2

These two equations can be solved simultaneously to find v1f and v2f. The algebraic manipulation can be complex, but it yields the following results:

v1f = ((m1 - m2) / (m1 + m2)) * v1i + ((2m2) / (m1 + m2)) * v2i

v2f = ((2m1) / (m1 + m2)) * v1i + ((m2 - m1) / (m1 + m2)) * v2i

These equations allow for the direct calculation of the final velocities of the objects involved in an elastic collision, given their initial velocities and masses.

Special Cases

Equal Masses

If the two objects have equal masses (m1 = m2 = m), the equations simplify to:

v1f = v2i

v2f = v1i

This means that the objects exchange velocities during the collision.

One Object Initially at Rest

If one object (say, m2) is initially at rest (v2i = 0), the equations become:

v1f = ((m1 - m2) / (m1 + m2)) * v1i

v2f = ((2m1) / (m1 + m2)) * v1i

These simplified equations are useful in various scenarios, such as analyzing collisions in a physics lab.

Example Problem

Consider two billiard balls. Ball 1 (m1 = 0.17 kg) is moving at 3 m/s, and it collides head-on with Ball 2 (m2 = 0.17 kg), which is at rest. Assuming the collision is perfectly elastic, find the final velocities of both balls.

Solution

Since m1 = m2, the balls exchange velocities. Therefore:

v1f = v2i = 0 m/s

v2f = v1i = 3 m/s

After the collision, Ball 1 stops, and Ball 2 moves with the initial velocity of Ball 1.

Advanced Concepts and Applications

Elastic collisions serve as a foundation for understanding more complex phenomena in physics.

Coefficient of Restitution

The coefficient of restitution (e) is a measure of how elastic a collision is. It is defined as the ratio of the relative velocity of separation after the collision to the relative velocity of approach before the collision:

e = - (v2f - v1f) / (v2i - v1i)

For perfectly elastic collisions, e = 1. For perfectly inelastic collisions, e = 0. Real-world collisions have values of e between 0 and 1.

Applications in Particle Physics

In particle physics, collisions between particles are fundamental to probing the structure of matter. Elastic collisions are essential in experiments where scientists study the interactions between elementary particles, such as electrons, protons, and neutrons. Analyzing the angles and energies of the scattered particles provides insights into the forces governing their interactions.

Molecular Dynamics Simulations

Elastic collisions are also used in molecular dynamics simulations to model the behavior of atoms and molecules. These simulations are crucial in fields such as chemistry, materials science, and biology, where understanding the interactions between atoms and molecules at the microscopic level is vital.

Limitations and Considerations

While the concept of elastic collisions is invaluable, it is essential to recognize its limitations.

Idealization

Perfectly elastic collisions are an idealization. In reality, some energy is always lost due to factors like friction, air resistance, or deformation of the colliding objects.

Macroscopic vs. Microscopic

At the macroscopic level, true elastic collisions are rare. However, at the microscopic level, collisions between atoms and molecules can often be approximated as elastic under certain conditions.

External Forces

The conservation laws apply to closed systems where no external forces act. If external forces are present, the total momentum and kinetic energy of the system will not be conserved.

Conclusion

In an elastic collision, both momentum and kinetic energy are conserved. This conservation allows for the detailed analysis and prediction of the outcomes of such collisions. While perfectly elastic collisions are an idealization, they provide a crucial framework for understanding a wide range of physical phenomena, from everyday scenarios like billiard balls to complex interactions in particle physics and molecular dynamics. Recognizing the principles and limitations of elastic collisions enhances our understanding of the fundamental laws governing the physical world.