What Is Conserved In Inelastic Collision

Inelastic collisions, unlike their elastic counterparts, are collisions where kinetic energy is not conserved. This means that some of the initial kinetic energy is converted into other forms of energy, such as thermal energy, sound, or deformation of the colliding objects. However, despite the loss of kinetic energy, certain fundamental physical quantities are always conserved in inelastic collisions. Understanding what is conserved is crucial for analyzing and predicting the outcomes of these collisions.

Conservation Principles in Inelastic Collisions

While kinetic energy takes a hit in inelastic collisions, the laws of physics dictate that certain quantities remain unchanged. These conserved quantities are the cornerstones for analyzing these collisions and include:

1. Momentum: The total momentum of the system remains constant.
2. Total Energy: The total energy of the system (including all forms of energy) is conserved.
3. Angular Momentum: The total angular momentum of the system is conserved.
4. Charge: The total electric charge is conserved.
5. Baryon Number and Lepton Number: In particle physics, these numbers are also conserved.

Let's delve into each of these conserved quantities and explore their implications in the context of inelastic collisions.

1. Conservation of Momentum

The law of conservation of momentum is a fundamental principle in physics, stating that the total momentum of a closed system remains constant if no external forces act on it. In simpler terms, this means that in a collision, the total momentum before the collision is equal to the total momentum after the collision.

Mathematical Representation:

The momentum (p) of an object is defined as the product of its mass (m) and velocity (v):

p = mv

For a system of multiple objects, the total momentum (P) is the vector sum of the individual momenta:

P = p1 + p2 + p3 + ... = m1v1 + m2v2 + m3v3 + ...

Therefore, the conservation of momentum can be expressed as:

P_initial = P_final

m1v1i + m2v2i = m1v1f + m2v2f

Where:

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

Implications for Inelastic Collisions:

Even though kinetic energy is lost in an inelastic collision, the total momentum of the system remains the same. This allows us to relate the initial and final velocities of the colliding objects, even if we don't know the exact amount of energy lost.

Example:

Consider a car (mass m1) crashing into a stationary truck (mass m2). The collision is inelastic because the car crumples and generates heat and sound. However, we can still use the conservation of momentum to determine the velocity of the combined car and truck immediately after the collision:

m1v1i + m2(0) = (m1 + m2)vf

vf = (m1v1i) / (m1 + m2)

Where:

• v1i is the initial velocity of the car.
• vf is the final velocity of the combined car and truck.

This equation allows us to calculate the final velocity without knowing how much kinetic energy was lost in the collision.

2. Conservation of Total Energy

The law of conservation of energy states that the total energy of an isolated system remains constant; energy can neither be created nor destroyed but can transform from one form to another.

Mathematical Representation:

The total energy (E) of a system is the sum of all forms of energy present, including:

• Kinetic Energy (KE)
• Potential Energy (PE)
• Thermal Energy (Q)
• Other forms of energy

Therefore, the conservation of energy can be expressed as:

E_initial = E_final

KEi + PEi + Qi + ... = KEf + PEf + Qf + ...

Implications for Inelastic Collisions:

In an inelastic collision, kinetic energy is converted into other forms of energy, such as thermal energy (heat), sound energy, and deformation energy. While kinetic energy is not conserved, the total energy of the system is conserved. This means that the initial kinetic energy, plus any initial potential energy, is equal to the final kinetic energy, plus the final potential energy, plus the energy converted into other forms.

Example:

Consider dropping a ball of clay onto the floor. The collision is highly inelastic because the clay deforms significantly and doesn't bounce back.

• Initially, the clay has potential energy (PEi) due to its height and kinetic energy (KEi) just before impact.
• After the collision, the clay has no kinetic energy (KEf = 0) and very little potential energy (PEf ≈ 0).
• Most of the initial energy is converted into thermal energy (Q) due to the deformation of the clay and the friction between the clay and the floor.

The conservation of energy tells us:

PEi + KEi = Qf

This shows that the initial potential and kinetic energy of the clay is entirely converted into thermal energy.

3. Conservation of Angular Momentum

Angular momentum (L) is a measure of an object's tendency to rotate. It is the rotational equivalent of linear momentum. The law of conservation of angular momentum states that the total angular momentum of a closed system remains constant if no external torque acts on it.

Mathematical Representation:

The angular momentum (L) of a point mass is defined as:

L = r x p = r x mv

Where:

• r is the position vector from the axis of rotation to the object.
• p is the linear momentum of the object.
• x represents the cross product.

For a rigid body rotating about a fixed axis, the angular momentum is:

L = Iω

Where:

• I is the moment of inertia of the object.
• ω is the angular velocity of the object.

The conservation of angular momentum can be expressed as:

L_initial = L_final

I1ω1i + I2ω2i = I1ω1f + I2ω2f

Implications for Inelastic Collisions:

If the collision involves rotating objects or objects that acquire rotation as a result of the collision, the conservation of angular momentum plays a crucial role. Even if kinetic energy is lost, the total angular momentum of the system remains constant.

Example:

Imagine a spinning figure skater pulling their arms inward. This is analogous to an inelastic collision in the sense that kinetic energy is being redistributed (the skater speeds up their rotation). The skater's moment of inertia (I) decreases as they pull their arms in. Since angular momentum (L = Iω) is conserved, their angular velocity (ω) must increase to compensate. The internal forces exerted by the skater pulling in their arms do not constitute an external torque.

Another example is a bullet fired into a rotating object, such as a merry-go-round. The bullet colliding with the merry-go-round is an inelastic collision. While kinetic energy is lost (converted to heat and sound), the total angular momentum of the bullet-merry-go-round system remains constant. This allows us to calculate the final angular velocity of the merry-go-round after the bullet embeds itself.

4. Conservation of Charge

The law of conservation of electric charge states that the total electric charge in an isolated system never changes. Charge can neither be created nor destroyed, but it can be transferred from one object to another.

Mathematical Representation:

The total charge (Q) in a system is the algebraic sum of all the individual charges:

Q = q1 + q2 + q3 + ...

Therefore, the conservation of charge can be expressed as:

Q_initial = Q_final

Implications for Inelastic Collisions:

The conservation of charge is always valid, regardless of the type of collision. In an inelastic collision, the total electric charge of the colliding objects remains the same before and after the collision.

Example:

Consider two charged particles colliding. Even if the collision is inelastic and kinetic energy is lost, the total charge of the two particles will remain the same. If one particle has a charge of +q and the other has a charge of -q initially, the total charge is zero. After the collision, the total charge will still be zero, even if the particles stick together or break apart into other charged particles. The sum of the charges of all the resulting particles must still equal zero.

5. Conservation of Baryon Number and Lepton Number

These conservation laws are specific to particle physics and are relevant when considering collisions at extremely high energies, where particles can be created and destroyed.

• Baryon Number: Baryons are particles made up of three quarks (e.g., protons and neutrons). The baryon number is a quantum number that is conserved in all known particle interactions. Each baryon has a baryon number of +1, each anti-baryon has a baryon number of -1, and all other particles have a baryon number of 0. The total baryon number before a collision must equal the total baryon number after the collision.
• Lepton Number: Leptons are fundamental particles that do not experience the strong force (e.g., electrons, muons, and neutrinos). Each lepton has a lepton number of +1, each anti-lepton has a lepton number of -1, and all other particles have a lepton number of 0. Lepton number is also conserved. Furthermore, individual lepton family numbers (electron lepton number, muon lepton number, and tau lepton number) are also often conserved.

Implications for Inelastic Collisions (Particle Physics):

In high-energy collisions, new particles can be created. However, these conservation laws place constraints on the types of particles that can be produced. For example, if a collision starts with a baryon number of +1, the resulting particles must also have a total baryon number of +1. This ensures that the fundamental building blocks of matter are neither created nor destroyed, only rearranged.

Example:

A proton (baryon number +1) collides with another proton (baryon number +1). The collision can create new particles, such as a proton, an antiproton, and a neutral pion:

p + p → p + p + p + p̄ + π⁰

• Proton (p): Baryon number = +1
• Antiproton (p̄): Baryon number = -1
• Neutral Pion (π⁰): Baryon number = 0

The total baryon number before the collision is +2 (1 + 1). After the collision, the total baryon number is also +2 (1 + 1 + 1 - 1 + 0). The conservation of baryon number is satisfied.

Why Kinetic Energy is NOT Conserved

The key distinction between elastic and inelastic collisions lies in the conservation of kinetic energy. In elastic collisions, kinetic energy is conserved; in inelastic collisions, it is not. This difference arises from the nature of the forces involved and the way energy is transferred during the collision.

• Energy Dissipation: In inelastic collisions, some of the initial kinetic energy is converted into other forms of energy. This energy dissipation is often due to:

  • Heat: Friction between the colliding objects can generate heat.
  • Sound: The collision can produce sound waves.
  • Deformation: The objects may deform permanently, requiring energy to break bonds and rearrange the material structure.
  • Internal Excitation: The collision can excite internal vibrations or rotations within the molecules of the colliding objects.

• Non-Conservative Forces: Inelastic collisions often involve non-conservative forces, such as friction or plastic deformation. Non-conservative forces do work that depends on the path taken, meaning that the energy lost is not recoverable. In contrast, elastic collisions involve conservative forces, such as the electrostatic force between atoms, where energy is stored in a potential energy function and can be recovered.

Real-World Examples of Inelastic Collisions

Inelastic collisions are far more common in everyday life than perfectly elastic collisions. Here are some examples:

• Car Accidents: As mentioned earlier, car crashes are highly inelastic. The crumpling of the car body, the sound of the impact, and the heat generated all represent kinetic energy being converted into other forms.
• Dropping a Ball: When you drop a ball, it bounces a few times before coming to rest. Each bounce is an inelastic collision, with some kinetic energy lost to heat and sound with each impact. The bouncier the ball, the more elastic the collision (and the less energy lost).
• Hammering a Nail: When you hammer a nail into a piece of wood, the collision between the hammer and the nail is inelastic. The kinetic energy of the hammer is used to deform the nail and drive it into the wood, generating heat and sound in the process.
• Catching a Ball: When you catch a ball, your hand exerts a force to stop it. The collision between the ball and your hand is inelastic, as some of the ball's kinetic energy is converted into heat and deformation in your hand and the ball.
• A Meteorite Impact: When a meteorite strikes the Earth, the collision is extremely inelastic, releasing a tremendous amount of energy as heat, light, and seismic waves.

Applying Conservation Laws to Solve Problems

The conservation laws discussed above are powerful tools for analyzing inelastic collisions and solving related problems. Here's a general approach:

1. Identify the System: Define the system of objects involved in the collision.
2. Identify External Forces: Determine if any external forces are acting on the system. If so, the conservation laws may not be strictly applicable. However, if the external forces are small compared to the forces during the collision, the conservation laws can still be a good approximation.
3. Apply Conservation of Momentum: Use the conservation of momentum equation to relate the initial and final velocities of the objects.
4. Apply Conservation of Energy: Use the conservation of energy equation to account for all forms of energy in the system, including kinetic energy, potential energy, and other forms of energy (heat, sound, deformation).
5. Apply Conservation of Angular Momentum (if applicable): If the collision involves rotating objects or objects that acquire rotation, use the conservation of angular momentum equation.
6. Apply Conservation of Charge (if applicable): Ensure that the total charge before and after the collision remains the same.
7. Solve the Equations: Solve the resulting equations to determine the unknown quantities.

Conclusion

Inelastic collisions, while not conserving kinetic energy, are governed by fundamental conservation laws that allow us to analyze and understand these interactions. The conservation of momentum, total energy, angular momentum, and charge are crucial for predicting the outcomes of inelastic collisions in various scenarios, from everyday events to high-energy particle interactions. By applying these conservation principles, we can gain valuable insights into the behavior of physical systems and solve a wide range of problems related to collisions. Understanding these principles provides a solid foundation for further exploration of mechanics and other areas of physics.