In Inelastic Collision What Is Conserved

In an inelastic collision, the fundamental principle that holds true is the conservation of momentum. While kinetic energy isn't conserved due to its transformation into other forms of energy, the total momentum of a closed system remains constant before, during, and after the collision. Understanding this principle is crucial for analyzing various physical interactions, from car crashes to particle physics.

Understanding Inelastic Collisions

Inelastic collisions represent a class of interactions where kinetic energy is not conserved. This means that some of the initial kinetic energy of the colliding objects is converted into other forms of energy such as:

• Heat: Generated due to friction between the colliding surfaces.
• Sound: Produced by the impact of the collision.
• Deformation: Energy used to permanently change the shape of the objects involved.

A classic example of an inelastic collision is a car crash. The impact results in significant deformation of the vehicles, generates heat due to friction, and produces a loud sound. All these transformations dissipate kinetic energy, making the collision inelastic. Another example is dropping a ball of clay onto the floor. The clay deforms upon impact and doesn't bounce back, indicating that most of its kinetic energy has been converted into deformation and heat.

Distinguishing inelastic collisions from elastic collisions is essential. In elastic collisions, both momentum and kinetic energy are conserved. Ideal elastic collisions are rare in everyday scenarios but can be approximated in situations like collisions between billiard balls.

The Law of Conservation of Momentum

The law of conservation of momentum is a cornerstone of physics. It states that the total momentum of an isolated system remains constant if no external forces act on it. In simpler terms, momentum, which is the product of mass and velocity, is neither lost nor gained in a closed system.

Mathematically, the conservation of momentum can be expressed as:

p_initial = p_final

Where:

• p_initial is the total momentum of the system before the collision.
• p_final is the total momentum of the system after the collision.

For a system of two objects undergoing a collision, the equation expands to:

m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f

Where:

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

This equation indicates that the sum of the momenta of the two objects before the collision equals the sum of their momenta after the collision. This principle holds true regardless of whether the collision is elastic or inelastic.

Why Momentum is Conserved in Inelastic Collisions

Momentum conservation stems from Newton's laws of motion, particularly Newton's third law, which states that for every action, there is an equal and opposite reaction. During a collision, objects exert forces on each other. These forces are internal to the system and act in opposite directions. According to Newton's third law, these forces are equal in magnitude but opposite in direction.

Consider two cars colliding. Car A exerts a force on Car B, and simultaneously, Car B exerts an equal and opposite force on Car A. These forces cause changes in the momentum of each car. However, because the forces are equal and opposite, the total change in momentum of the system (both cars) is zero.

Mathematically, this can be represented as:

F₁ = -F₂

Where:

• F₁ is the force exerted by Car A on Car B.
• F₂ is the force exerted by Car B on Car A.

Since force is the rate of change of momentum (F = dp/dt), we can write:

dp₁/dt = -dp₂/dt

Integrating both sides with respect to time gives:

Δp₁ = -Δp₂

This means the change in momentum of Car A is equal and opposite to the change in momentum of Car B. Therefore, the total change in momentum of the system is zero, confirming the conservation of momentum.

Kinetic Energy Loss in Inelastic Collisions

Unlike momentum, kinetic energy is not conserved in inelastic collisions. Kinetic energy is the energy an object possesses due to its motion, given by the formula:

KE = 1/2 mv²

Where:

• KE is kinetic energy.
• m is mass.
• v is velocity.

In an inelastic collision, some of this kinetic energy is converted into other forms of energy, such as heat, sound, and deformation. The amount of kinetic energy lost can vary depending on the nature of the collision and the materials involved.

For example, consider a perfectly inelastic collision where two objects stick together after impact. This is a scenario where the maximum amount of kinetic energy is lost. Let's analyze a simple case:

Two objects with masses m₁ and m₂ are moving with initial velocities v₁ᵢ and v₂ᵢ, respectively. After the collision, they stick together and move with a common final velocity vf.

Using conservation of momentum:

m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)vf

Solving for vf:

vf = (m₁v₁ᵢ + m₂v₂ᵢ) / (m₁ + m₂)

The initial kinetic energy is:

KEᵢ = 1/2 m₁v₁ᵢ² + 1/2 m₂v₂ᵢ²

The final kinetic energy is:

KEf = 1/2 (m₁ + m₂)vf²

Substituting the expression for vf:

KEf = 1/2 (m₁ + m₂) [(m₁v₁ᵢ + m₂v₂ᵢ) / (m₁ + m₂)]²

KEf = 1/2 (m₁v₁ᵢ + m₂v₂ᵢ)² / (m₁ + m₂)

The change in kinetic energy (loss) is:

ΔKE = KEf - KEᵢ

ΔKE = 1/2 (m₁v₁ᵢ + m₂v₂ᵢ)² / (m₁ + m₂) - (1/2 m₁v₁ᵢ² + 1/2 m₂v₂ᵢ²)

This expression will always yield a negative value, indicating a loss of kinetic energy. The lost kinetic energy is transformed into other forms of energy, such as heat and deformation.

Types of Inelastic Collisions

Inelastic collisions can be further categorized based on the extent of kinetic energy loss and the behavior of the colliding objects:

1. Perfectly Inelastic Collisions: These collisions involve the maximum loss of kinetic energy. The colliding objects stick together and move as a single mass after the collision. Examples include:

   • A bullet embedding itself in a wooden block.
   • Two railway cars coupling together.
   • A blob of clay sticking to a wall.

2. Partially Inelastic Collisions: These collisions involve some loss of kinetic energy, but the objects do not stick together. They bounce off each other, but the total kinetic energy after the collision is less than before. Examples include:

   • A rubber ball bouncing on the floor (some energy is lost as heat and sound).
   • A car crash where the vehicles do not remain entangled.

Real-World Applications of Momentum Conservation in Inelastic Collisions

The principle of momentum conservation in inelastic collisions has numerous practical applications across various fields:

1. Automotive Safety: Car manufacturers use the principles of momentum and impulse to design safer vehicles. Crumple zones are designed to deform during a collision, increasing the collision time and reducing the force experienced by the occupants. Seatbelts and airbags further distribute the force of impact over a larger area and time, reducing the risk of injury.

2. Sports: In sports like football and hockey, understanding momentum conservation is crucial. Players use their mass and velocity to generate momentum, which they can transfer to other players or objects (like the ball or puck). Coaches use this knowledge to develop strategies that maximize the transfer of momentum and improve performance.

3. Ballistics: The study of projectile motion relies heavily on momentum conservation. When a bullet is fired from a gun, the momentum gained by the bullet is equal and opposite to the momentum gained by the gun (recoil). This principle is used to calculate the velocity and trajectory of projectiles.

4. Aerospace Engineering: In spacecraft docking, momentum conservation is critical. Spacecraft must carefully adjust their velocities to ensure a smooth and controlled docking process. Inelastic collisions can occur during docking, and engineers must account for the loss of kinetic energy and the conservation of momentum to maintain stability.

5. Particle Physics: In particle accelerators, physicists study collisions between subatomic particles to understand the fundamental laws of nature. While these collisions often involve high energies and relativistic effects, the principle of momentum conservation remains valid. By analyzing the momenta of the particles before and after the collision, physicists can infer the properties of new particles and interactions.

Examples and Calculations

To illustrate the application of momentum conservation in inelastic collisions, consider the following examples:

Example 1: Perfectly Inelastic Collision

A car with a mass of 1500 kg is moving at 20 m/s when it collides head-on with a stationary truck with a mass of 3000 kg. If the vehicles stick together after the collision, what is their final velocity?

• m₁ (car) = 1500 kg
• v₁ᵢ (car) = 20 m/s
• m₂ (truck) = 3000 kg
• v₂ᵢ (truck) = 0 m/s

Using the conservation of momentum:

m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)vf

(1500 kg)(20 m/s) + (3000 kg)(0 m/s) = (1500 kg + 3000 kg)vf

30000 kg·m/s = (4500 kg)vf

vf = 30000 kg·m/s / 4500 kg

vf ≈ 6.67 m/s

The final velocity of the combined vehicles is approximately 6.67 m/s in the direction of the car's initial motion.

Example 2: Partially Inelastic Collision

A bowling ball with a mass of 7 kg is rolling at 5 m/s when it strikes a pin with a mass of 1.5 kg. After the collision, the pin moves forward at 8 m/s. What is the final velocity of the bowling ball?

• m₁ (bowling ball) = 7 kg
• v₁ᵢ (bowling ball) = 5 m/s
• m₂ (pin) = 1.5 kg
• v₂ᵢ (pin) = 0 m/s
• v₂f (pin) = 8 m/s

Using the conservation of momentum:

m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f

(7 kg)(5 m/s) + (1.5 kg)(0 m/s) = (7 kg)v₁f + (1.5 kg)(8 m/s)

35 kg·m/s = (7 kg)v₁f + 12 kg·m/s

(7 kg)v₁f = 35 kg·m/s - 12 kg·m/s

(7 kg)v₁f = 23 kg·m/s

v₁f = 23 kg·m/s / 7 kg

v₁f ≈ 3.29 m/s

The final velocity of the bowling ball is approximately 3.29 m/s in the same direction as its initial motion.

Advanced Considerations

While the basic principle of momentum conservation is straightforward, some advanced considerations can affect its application:

1. External Forces: The conservation of momentum applies to closed systems where no external forces are acting. In real-world scenarios, external forces such as friction and air resistance may be present. These forces can affect the total momentum of the system and must be accounted for in more complex analyses.

2. Relativistic Effects: At very high speeds, approaching the speed of light, classical mechanics breaks down, and relativistic effects become significant. In these cases, the relativistic momentum, given by p = γmv (where γ is the Lorentz factor), must be used. The conservation of relativistic momentum still holds true in inelastic collisions.

3. Rotational Motion: In collisions involving extended objects, rotational motion and angular momentum may also be involved. The conservation of angular momentum must be considered in addition to linear momentum to fully analyze the collision.

4. Impulse: The concept of impulse is closely related to momentum conservation. Impulse is the change in momentum of an object, given by J = Δp = FΔt, where J is the impulse, F is the force, and Δt is the time interval over which the force acts. In collisions, the impulse experienced by each object is equal and opposite, leading to the conservation of momentum.

Common Misconceptions

Several common misconceptions surround the concept of momentum conservation in inelastic collisions:

1. Kinetic Energy is Always Conserved: This is incorrect. Kinetic energy is only conserved in elastic collisions. In inelastic collisions, some kinetic energy is always converted into other forms of energy.

2. Momentum is Only Conserved in Elastic Collisions: This is also incorrect. Momentum is conserved in all collisions, regardless of whether they are elastic or inelastic, provided that the system is closed and no external forces are acting.

3. Heavier Objects Always Retain More Momentum: While heavier objects have more momentum at the same velocity, the distribution of momentum after a collision depends on the initial momenta of all objects involved, not just their masses.

4. Collisions Always Result in a Complete Stop: This is not true. In many inelastic collisions, the objects continue to move after the collision, albeit with different velocities. Only in perfectly inelastic collisions where objects stick together might the final velocity be zero if the initial momenta cancel out.

Conclusion

In summary, while kinetic energy is not conserved in inelastic collisions, the principle of conservation of momentum remains a fundamental and unwavering law. This principle, derived from Newton's laws of motion, states that the total momentum of a closed system remains constant before, during, and after a collision. Understanding and applying this principle is crucial for analyzing a wide range of physical phenomena, from car crashes to particle interactions. By mastering the concepts of momentum, impulse, and kinetic energy loss, one can gain a deeper understanding of the dynamics of collisions and their real-world applications.