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Inelastic Collision

Updated Jun 14, 2026

An inelastic collision is a collision in which the total kinetic energy of the colliding bodies is not conserved. While the total momentum of the system remains constant—provided no external forces act upon the system—the internal kinetic energy is transformed into other forms of energy, such as thermal energy, sound, or permanent deformation of the objects involved.

On This Page

  1. Short Definition
  2. Detailed Explanation
  3. Key Formula / Syntax / Principle
  4. Important Characteristics
  5. Practical Example
  6. Common Confusions or Misconceptions
  7. Related Terms
  8. Why It Matters

Short Definition

In physics, an inelastic collision occurs when the total kinetic energy of a system changes during a collision. Unlike an elastic collision, where kinetic energy is perfectly preserved, an inelastic collision involves the dissipation of energy. Despite this loss of kinetic energy, the law of conservation of momentum holds true, meaning the total momentum of the objects before the collision is equal to the total momentum after the collision.

Detailed Explanation

In Newtonian mechanics, collisions are categorized by the conservation of kinetic energy. In an inelastic collision, internal forces acting during the impact cause a portion of the kinetic energy to be converted into non-mechanical forms. This energy transformation typically occurs due to friction, air resistance, or internal structural changes such as crushing or bending.

The degree of "inelasticity" is measured using the coefficient of restitution (##e##). This scalar value ranges from 0 to 1. If ##e = 1##, the collision is perfectly elastic. If ##0 \le e < 1##, the collision is inelastic. When ##e = 0##, the collision is described as "perfectly inelastic," meaning the objects stick together after the impact and move with a common final velocity.

During the collision process, the objects experience impulsive forces. These forces act over a very short duration, causing rapid changes in velocity. Because the collision is internal to the system, Newton's Third Law dictates that the force exerted by object A on object B is equal and opposite to the force exerted by B on A, ensuring that the net impulse on the system is zero.

Key Formula / Syntax / Principle

The primary governing principle for all collisions is the Conservation of Linear Momentum. For two objects with masses ##m_1## and ##m_2## and initial velocities ##u_1## and ##u_2##:

###m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2###

Where ##v_1## and ##v_2## are the final velocities. In a perfectly inelastic collision, the objects merge:

###m_1u_1 + m_2u_2 = (m_1 + m_2)v_f###

The coefficient of restitution (##e##) relates the relative velocities before and after the collision:

###e = \dfrac{v_2 - v_1}{u_1 - u_2}###
Collision Type Kinetic Energy Coefficient (##e##)
Perfectly Elastic Conserved ##e = 1##
Inelastic Not Conserved ##0 < e < 1##
Perfectly Inelastic Maximum Loss ##e = 0##

Important Characteristics

The defining characteristic of an inelastic collision is the permanent change in the state of the objects or the surrounding environment.

Practical Example

A common example of an inelastic collision is a ballistic pendulum or a car crash. In a vehicle collision involving two cars that crumple upon impact, the kinetic energy of the vehicles is used to deform the metal frames. Because the metal does not return to its original shape, that energy is lost from the system's kinetic energy budget.

Consider a 1000 kg car traveling at 20 m/s hitting a stationary 1000 kg car. If they stick together (perfectly inelastic), the conservation of momentum dictates:

###(1000 \times 20) + (1000 \times 0) = (1000 + 1000) \times v_f###
###20,000 = 2000 \times v_f###
###v_f = 10 \text{ m/s}###

The final velocity is 10 m/s. The initial kinetic energy was ##200,000\text{ J}##, while the final kinetic energy is ##100,000\text{ J}##. The ##100,000\text{ J}## difference represents the energy dissipated through the deformation of the vehicles.

Factor Elastic Collision Inelastic Collision
Total Momentum Conserved Conserved
Kinetic Energy Conserved Not Conserved
Internal Energy Constant Increased

Common Confusions or Misconceptions

A frequent misconception is that "inelastic" implies that momentum is lost. This is incorrect. Momentum is a vector quantity that remains conserved in all collisions where no external horizontal forces (like friction from the road or air drag) significantly alter the system's state during the split-second of impact.

Another point of confusion is the assumption that objects must "stick together" to be inelastic. While "perfectly inelastic" requires objects to move with a common velocity, any collision where kinetic energy is not conserved—even if the objects bounce off each other—is classified as inelastic.

Why It Matters

Understanding inelastic collisions is critical for engineering safety systems. Automotive designers use the principles of inelastic collisions to design "crumple zones" in vehicles. By intentionally making the front of a car deform in an inelastic manner, engineers ensure that a significant portion of the kinetic energy is absorbed by the vehicle's structure rather than by the passengers, thereby reducing the force of impact on the occupants. This principle is applied in everything from sports equipment design to planetary formation models in astrophysics.

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