What Is Conserved In An Inelastic Collision

Inelastic collisions are ubiquitous in our everyday lives, from car crashes to the simple act of bouncing a ball off the floor. Unlike their perfectly elastic counterparts, inelastic collisions involve a transformation of kinetic energy into other forms of energy, such as heat, sound, or deformation. Understanding what remains constant, or is conserved, during these collisions is crucial for analyzing and predicting their outcomes.

Understanding Inelastic Collisions

Before diving into the conserved quantities, it's important to define what makes a collision inelastic. An inelastic collision is one where the total kinetic energy of the system is not conserved. This means that some of the initial kinetic energy is converted into other forms of energy.

Think of a car crash: the kinetic energy of the vehicles is converted into the energy required to crumple metal, produce heat from friction, and generate sound. A ball of clay dropped on the floor provides another example; it deforms upon impact, and the kinetic energy is used to change its shape.

Here are some key characteristics of inelastic collisions:

• Kinetic Energy is Not Conserved: This is the defining feature. The total kinetic energy before the collision is greater than the total kinetic energy after the collision.
• Energy Transformation: The "lost" kinetic energy is converted into other forms of energy.
• Objects May Stick Together: In some inelastic collisions, the colliding objects stick together and move as a single mass after the impact. These are called perfectly inelastic collisions.

What is Conserved in Inelastic Collisions?

Despite the loss of kinetic energy, certain fundamental quantities are conserved in all collisions, including inelastic ones. These are:

1. Total Energy: This is the most fundamental conservation law. The total energy of a closed system always remains constant. Energy can be transformed from one form to another, but it cannot be created or destroyed.
2. Momentum: The total momentum of a closed system is conserved in all collisions, elastic or inelastic.
3. Total Mass: In non-relativistic collisions, the total mass of the system remains constant.
4. Charge: The total electric charge is always conserved.

Let's delve deeper into each of these conserved quantities.

1. Conservation of Total Energy

The principle of energy conservation is a cornerstone of physics. It dictates that the total energy within a closed system remains constant over time. In the context of an inelastic collision, this means that while kinetic energy is not conserved, the total energy is. The kinetic energy that disappears is transformed into other forms of energy, ensuring that the total energy of the system remains the same.

Mathematically, this can be represented as:

E_{total (initial)} = E_{total (final)}

Where:

• E_{total (initial)} is the total energy of the system before the collision.
• E_{total (final)} is the total energy of the system after the collision.

This total energy includes all forms of energy, such as:

• Kinetic energy
• Potential energy
• Thermal energy (heat)
• Sound energy
• Energy stored in deformation

Consider the car crash example again. The initial kinetic energy of the cars is converted into:

• Heat due to friction between the colliding parts.
• Sound energy from the loud crash.
• Energy used to deform the metal of the cars.
• Possibly, potential energy if the cars are moved to a different height.

If you were to measure all these forms of energy after the collision and sum them up, you would find that the total energy is equal to the initial kinetic energy of the cars (plus any other initial forms of energy).

2. Conservation of Momentum

Momentum, denoted by p, is a measure of an object's mass in motion. It is calculated as the product of an object's mass (m) and its velocity (v):

p = mv

Momentum is a vector quantity, meaning it has both magnitude and direction. The law of conservation of momentum states that the total momentum of a closed system remains constant in the absence of external forces. A closed system is one where no external forces act on the objects within the system.

Mathematically, the conservation of momentum can be expressed as:

p_{total (initial)} = p_{total (final)}

Where:

• p_{total (initial)} is the vector sum of the momenta of all objects in the system before the collision.
• p_{total (final)} is the vector sum of the momenta of all objects in the system after the collision.

Let's break this down with an example:

Imagine two objects colliding head-on. Object A has a mass m_A and an initial velocity v_Ai, and Object B has a mass m_B and an initial velocity v_Bi. After the collision, their velocities are v_Af and v_Bf, respectively.

The conservation of momentum equation would look like this:

m_Av_Ai + m_Bv_Bi = m_Av_Af + m_Bv_Bf

This equation tells us that the total momentum of the two objects before the collision is equal to the total momentum after the collision.

Even in a perfectly inelastic collision, where the objects stick together after the impact, momentum is still conserved. In this case, the final velocity of the combined mass (v_f) can be calculated as:

m_Av_Ai + m_Bv_Bi = (m_A + m_B)v_f

Why is Momentum Conserved?

The conservation of momentum is a direct consequence of Newton's Third Law of Motion: For every action, there is an equal and opposite reaction. During a collision, the objects exert forces on each other. These forces are equal in magnitude and opposite in direction. According to Newton's Second Law (F = ma), force is related to the change in momentum. Since the forces are equal and opposite, the changes in momentum are also equal and opposite, resulting in no net change in the total momentum of the system.

3. Conservation of Total Mass

In classical physics, the law of conservation of mass states that mass cannot be created or destroyed in a closed system. This means that the total mass of the system before a collision is equal to the total mass after the collision.

m_{total (initial)} = m_{total (final)}

This principle holds true for non-relativistic collisions, where the speeds of the objects are much smaller than the speed of light. However, at relativistic speeds (approaching the speed of light), mass and energy are interchangeable according to Einstein's famous equation E=mc². In such cases, we must consider the conservation of mass-energy, which is a more general principle. For most everyday collisions, the change in mass due to energy conversion is negligible.

Consider our car crash example again. The combined mass of the cars before the collision is (to a very, very close approximation) equal to the combined mass of the wreckage after the collision.

4. Conservation of Charge

Electric charge is a fundamental property of matter, and it is always conserved. The law of conservation of charge states that the total electric charge in a closed system remains constant. Charge can be transferred from one object to another, but the net charge of the system remains the same.

While not always immediately apparent in macroscopic collisions, the conservation of charge is crucial at the atomic and subatomic levels. For instance, when atoms collide and exchange electrons, the total charge of the system remains constant.

Applying Conservation Laws to Solve Problems

The conservation laws provide powerful tools for analyzing and solving problems involving inelastic collisions. By applying these principles, we can determine unknown quantities such as final velocities, energy losses, or the forces involved in the collision.

Here's a general approach to solving inelastic collision problems:

1. Define the System: Identify the objects involved in the collision and define the system. Ensure that the system is closed, meaning no external forces are acting on it (or that the effects of external forces are negligible).
2. Identify Knowns and Unknowns: List the known quantities (masses, initial velocities, etc.) and the unknowns you need to determine.
3. Apply Conservation Laws:
   • Momentum: Set up the conservation of momentum equation: p_{total (initial)} = p_{total (final)}.
   • Energy: Remember that kinetic energy is not conserved in inelastic collisions. However, total energy is conserved. If you need to analyze energy transformations, consider the different forms of energy involved.
   • Mass: In non-relativistic collisions, use the conservation of mass: m_{total (initial)} = m_{total (final)}.
   • Charge: Make sure the total charge is conserved.
4. Solve the Equations: Solve the equations you've set up to find the unknown quantities. Remember that momentum is a vector, so you may need to break down the problem into components.
5. Interpret the Results: Check if your answers make sense physically. For example, are the final velocities reasonable? Is the energy loss consistent with the type of collision?

Examples of Inelastic Collisions

Here are some real-world examples of inelastic collisions:

• Car Accidents: As discussed earlier, car crashes are classic examples of inelastic collisions. Kinetic energy is converted into heat, sound, and deformation of the vehicles.
• Dropping a Ball of Clay: When a ball of clay is dropped on the floor, it deforms upon impact. The kinetic energy is used to change the shape of the clay, and very little energy is returned as rebound.
• A Bullet Hitting a Target: When a bullet strikes a target, it embeds itself into the target. The kinetic energy of the bullet is converted into heat and deformation of both the bullet and the target. This is a perfectly inelastic collision.
• A Bat Hitting a Baseball: While a baseball bat hitting a baseball may seem close to elastic, it is actually slightly inelastic. Some of the kinetic energy is lost as heat and sound during the impact.
• Explosions (Reverse Collision): While not a collision in the traditional sense, explosions can be thought of as the reverse of a perfectly inelastic collision. A single object (e.g., a bomb) initially at rest breaks apart into multiple fragments, with kinetic energy being released. The total momentum of the fragments still sums to zero (the initial momentum of the bomb).

Inelastic Collisions in Different Frames of Reference

The description of an inelastic collision can vary depending on the frame of reference of the observer. While the conservation laws hold true in all inertial frames of reference, the observed values of kinetic energy and momentum may be different.

For example, consider a collision between two objects observed from two different frames:

• Frame 1: An observer at rest relative to the center of mass of the two objects. In this frame, the total momentum of the system is zero.
• Frame 2: An observer moving at a constant velocity relative to the center of mass. In this frame, the total momentum of the system is non-zero.

While the numerical values of momentum and kinetic energy will be different in the two frames, the fundamental conservation laws still apply. The total momentum in each frame remains constant before and after the collision, and the total energy (including all forms) is also conserved in each frame.

Elastic vs. Inelastic Collisions: A Comparison

To solidify your understanding, let's compare elastic and inelastic collisions:

Conclusion

Inelastic collisions are a fundamental part of our physical world. While kinetic energy is not conserved in these interactions, the total energy, momentum, mass (in non-relativistic scenarios), and charge are conserved. Understanding these conservation laws allows us to analyze and predict the outcomes of inelastic collisions, providing valuable insights into a wide range of phenomena from car accidents to atomic interactions. By applying these principles, we can gain a deeper understanding of the fundamental laws that govern our universe.