In Inelastic Collision What Is Conserved

An inelastic collision, a concept fundamental to physics, describes a collision where kinetic energy isn't conserved. However, grasping what remains constant during such an event is crucial for understanding momentum, energy transfer, and the behavior of systems in motion.

Understanding Inelastic Collisions

Inelastic collisions stand in contrast to elastic collisions, where kinetic energy is conserved. In the real world, perfectly elastic collisions are rare; most collisions fall into the inelastic category. Think of a car crash, a ball of clay hitting the floor, or even the simple act of two billiard balls colliding (though billiard balls are designed to approximate elastic collisions).

The defining characteristic of an inelastic collision is the conversion of kinetic energy into other forms of energy. This could be:

• Heat: Friction between colliding objects generates heat.
• Sound: The impact produces sound waves.
• Deformation: The objects may change shape permanently.
• Internal Energy: The internal energy of the colliding objects might increase, manifesting as increased molecular vibration.

Despite the loss of kinetic energy, a crucial principle remains steadfast: the law of conservation of momentum.

The Law of Conservation of Momentum

The law of conservation of momentum is a cornerstone of physics, stating that the total momentum of a closed system remains constant if no external forces act on it. A closed system means no mass enters or leaves the system, and no external forces like friction or air resistance are present. In simpler terms, in a collision, the total momentum before the collision equals the total momentum after the collision.

Momentum (p) is defined as the product of an object's mass (m) and its velocity (v):

p = mv

For a system of multiple objects, the total momentum is the vector sum of the individual momenta. Therefore, for a two-object collision:

p_1i + p_2i = p_1f + p_2f

Where:

• p_1i is the initial momentum of object 1.
• p_2i is the initial momentum of object 2.
• p_1f is the final momentum of object 1.
• p_2f is the final momentum of object 2.

This equation holds true regardless of whether the collision is elastic or inelastic. The conservation of momentum is a fundamental principle rooted in Newton's laws of motion, particularly the law of action and reaction.

Why Momentum is Conserved: Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. During a collision, object 1 exerts a force on object 2, and simultaneously, object 2 exerts an equal and opposite force on object 1. These forces act for the same duration of time.

Since Impulse (J), which is the change in momentum, is defined as the force (F) multiplied by the time interval (Δt) over which the force acts:

J = FΔt = Δp

The impulse experienced by each object is equal in magnitude but opposite in direction. Therefore, the change in momentum of object 1 is equal and opposite to the change in momentum of object 2. This means the total change in momentum of the system is zero, and the total momentum remains constant.

Types of Inelastic Collisions

Inelastic collisions are not all created equal. They can be further categorized based on the degree to which kinetic energy is lost:

1. Perfectly Inelastic Collisions: In a perfectly inelastic collision, the objects stick together after the impact, moving as a single combined mass. This represents the maximum loss of kinetic energy. A classic example is a bullet embedding itself in a block of wood. The final velocity of the combined mass can be calculated using the conservation of momentum:

   m₁v_1i + m₂v_2i = (m₁ + m₂)v_f

   Where:

   • v_f is the final velocity of the combined mass.

2. Partially Inelastic Collisions: These collisions fall between elastic and perfectly inelastic collisions. Kinetic energy is lost, but the objects do not stick together. Car crashes, where the vehicles deform but continue moving separately, are often considered partially inelastic. Analyzing these collisions requires knowing the coefficient of restitution (e), which is a measure of the "bounciness" of the collision. It's defined as the ratio of the relative velocity of separation to the relative velocity of approach:

   e = -(v_2f - v_1f) / (v_2i - v_1i)

   • e = 1 for perfectly elastic collisions.
   • e = 0 for perfectly inelastic collisions.
   • 0 < e < 1 for partially inelastic collisions.

   Knowing the coefficient of restitution, along with the conservation of momentum, allows you to solve for the final velocities of the objects.

Mathematical Examples and Problem-Solving

Let's illustrate the conservation of momentum in inelastic collisions with a few examples:

Example 1: Perfectly Inelastic Collision

A 5 kg bowling ball moving at 8 m/s strikes a 1 kg bowling pin initially at rest. The ball and pin stick together after the collision. What is their final velocity?

• m₁ = 5 kg (bowling ball)
• v_1i = 8 m/s (initial velocity of bowling ball)
• m₂ = 1 kg (bowling pin)
• v_2i = 0 m/s (initial velocity of bowling pin)

Using the conservation of momentum equation for a perfectly inelastic collision:

m₁v_1i + m₂v_2i = (m₁ + m₂)v_f

(5 kg)(8 m/s) + (1 kg)(0 m/s) = (5 kg + 1 kg)v_f

40 kg m/s = (6 kg)v_f

v_f = 40 kg m/s / 6 kg = 6.67 m/s

Therefore, the final velocity of the bowling ball and pin together is 6.67 m/s.

Example 2: Partially Inelastic Collision

Two cars collide head-on. Car A has a mass of 1500 kg and is traveling at 20 m/s eastward. Car B has a mass of 1200 kg and is traveling at 15 m/s westward. The coefficient of restitution for the collision is 0.4. What are the final velocities of the two cars?

• m_A = 1500 kg
• v_Ai = 20 m/s (eastward - positive direction)
• m_B = 1200 kg
• v_Bi = -15 m/s (westward - negative direction)
• e = 0.4

We have two equations: conservation of momentum and the coefficient of restitution equation:

1. m_Av_Ai + m_Bv_Bi = m_Av_Af + m_Bv_Bf
2. e = -(v_Bf - v_Af) / (v_Bi - v_Ai)

Substituting the given values:

1. (1500 kg)(20 m/s) + (1200 kg)(-15 m/s) = (1500 kg)v_Af + (1200 kg)v_Bf 30000 kg m/s - 18000 kg m/s = 1500v_Af + 1200v_Bf 12000 = 1500v_Af + 1200v_Bf
2. 0. 4 = -(v_Bf - v_Af) / (-15 m/s - 20 m/s)
   1. 4 = -(v_Bf - v_Af) / (-35 m/s)
   2. 4 * -35 = -v_Bf + v_Af -14 = -v_Bf + v_Af v_Af = v_Bf - 14

Now substitute the value of v_Af in equation 1:

12000 = 1500(v_Bf - 14) + 1200v_Bf 12000 = 1500v_Bf - 21000 + 1200v_Bf 33000 = 2700v_Bf v_Bf = 33000 / 2700 = 12.22 m/s

Now find v_Af:

v_Af = v_Bf - 14 v_Af = 12.22 - 14 = -1.78 m/s

Therefore, the final velocity of car A is -1.78 m/s (westward), and the final velocity of car B is 12.22 m/s (eastward).

Applications of Inelastic Collisions

The principles of inelastic collisions are applied in various fields:

• Vehicle Safety: Car manufacturers use the understanding of inelastic collisions to design safer vehicles. Crumple zones are designed to deform during a crash, absorbing kinetic energy and reducing the force of impact on the occupants.
• Sports: Understanding collisions is crucial in sports like baseball, football, and hockey. The design of equipment like helmets and padding aims to minimize the impact forces experienced by athletes during collisions.
• Ballistics: The study of projectile motion and impact relies heavily on the principles of inelastic collisions, particularly in analyzing the impact of bullets on targets.
• Industrial Processes: Many industrial processes, such as forging and hammering, involve inelastic collisions to shape materials.
• Astrophysics: Inelastic collisions play a role in the formation of planets and the evolution of planetary rings, where dust particles collide and coalesce.

Common Misconceptions

• Inelastic collisions mean objects always stick together: While perfectly inelastic collisions involve objects sticking together, partially inelastic collisions do not. The objects separate after the collision, albeit with a loss of kinetic energy.
• Energy is completely lost in an inelastic collision: Energy isn't "lost" but transformed into other forms like heat, sound, and deformation. The total energy of the system remains conserved, as dictated by the law of conservation of energy. However, the kinetic energy is reduced.
• Momentum is only conserved in elastic collisions: This is a crucial misconception. Momentum is conserved in all collisions, regardless of whether they are elastic or inelastic, as long as the system is closed (no external forces).

The Role of Internal Forces

While external forces violate the conservation of momentum, internal forces within the system don't. The forces between the colliding objects are internal forces. They contribute to the transfer of momentum between the objects but do not change the total momentum of the system.

Friction during the collision is an internal force. It converts kinetic energy into heat, making the collision inelastic. However, it doesn't affect the overall conservation of momentum of the system composed of the colliding objects. If, however, friction with the ground acted on the objects, that would be an external force, and momentum would not be conserved for the system of just the two colliding objects. You would need to include the Earth in your system to account for that external force.

Advanced Considerations: Systems with Variable Mass

The simple momentum conservation equations discussed above assume that the mass of the system remains constant. However, there are situations where the mass changes during the collision or interaction. Examples include:

• Rockets: Rockets expel mass (exhaust gases) to generate thrust. The mass of the rocket decreases over time.
• Conveyor Belts: If sand is dropped onto a moving conveyor belt, the mass of the belt-plus-sand system increases.

In these cases, the basic p = mv equation isn't sufficient. A more general formulation of momentum conservation is required, involving calculus and considering the rate of change of mass. The rocket equation, for example, is derived from the principle of momentum conservation applied to a system with variable mass.

Conservation of Energy Revisited

Although kinetic energy is not conserved in an inelastic collision, the total energy of the system is always conserved, in accordance with the law of conservation of energy. The lost kinetic energy is simply transformed into other forms of energy. To account for this energy transformation, we can write:

KE_i = KE_f + Q

Where:

• KE_i is the initial kinetic energy of the system.
• KE_f is the final kinetic energy of the system.
• Q is the energy converted into other forms (heat, sound, deformation, etc.).

Calculating the value of Q can be complex, as it requires knowing the details of the energy transformation processes. In many practical problems, we focus on momentum conservation to determine the final velocities and then calculate the change in kinetic energy to understand how much energy was converted.

Examples in Different Frames of Reference

The principle of conservation of momentum holds true in all inertial frames of reference (frames that are not accelerating). However, the values of the momenta and velocities will differ depending on the chosen frame.

For example, consider two cars colliding. An observer standing on the side of the road will measure different velocities than an observer moving in another car. However, both observers will agree that the total momentum of the two-car system is conserved in their respective frame of reference.

This illustrates the relativity of motion. While the specific numbers depend on the observer, the fundamental laws of physics, including the conservation of momentum, remain invariant.

Conclusion

Inelastic collisions, characterized by the loss of kinetic energy, are ubiquitous in the real world. While kinetic energy is not conserved, the law of conservation of momentum remains a steadfast principle. This principle, rooted in Newton's laws of motion, allows us to analyze and predict the motion of objects after a collision, even when energy is transformed into other forms. Understanding inelastic collisions is crucial for diverse fields, from designing safer vehicles to analyzing astrophysical phenomena. Recognizing the nuances of different types of inelastic collisions, applying the correct equations, and avoiding common misconceptions are essential for mastering this fundamental concept in physics. By consistently applying the law of conservation of momentum, we can unlock a deeper understanding of how objects interact and move in our world.