Types of collisions — elastic, inelastic, perfectly inelastic with conservation laws

Question

Compare elastic, inelastic, and perfectly inelastic collisions. Which conservation laws apply in each case? What is the coefficient of restitution, and how does it classify collisions?

(JEE Main 2024 asked a perfectly inelastic collision problem; NEET tests conservation law application)

Solution — Step by Step

Momentum is ALWAYS conserved in all types of collisions (assuming no external forces).

m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2

This is non-negotiable. Whether elastic or inelastic, momentum conservation always applies.

In a perfectly elastic collision, both momentum AND kinetic energy are conserved.

\frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2

For a 1D elastic collision, the velocity exchange formula: $v_1 = \frac{(m_1-m_2)u_1 + 2m_2u_2}{m_1+m_2}$

Special case: equal masses → velocities exchange completely.

The objects merge into one after collision. Maximum kinetic energy is lost (converted to heat, sound, deformation).

m_1u_1 + m_2u_2 = (m_1 + m_2)v

v = \frac{m_1u_1 + m_2u_2}{m_1 + m_2}

KE loss: $\Delta KE = \frac{1}{2}\frac{m_1 m_2}{m_1 + m_2}(u_1 - u_2)^2$

e = \frac{\text{relative velocity of separation}}{\text{relative velocity of approach}} = \frac{v_2 - v_1}{u_1 - u_2}

$e = 1$ : perfectly elastic
$0 < e < 1$ : inelastic (most real collisions)
$e = 0$ : perfectly inelastic (objects stick)

flowchart TD
    A[Collision occurs] --> B{Coefficient of restitution e?}
    B -->|e = 1| C["Elastic<br/>KE conserved<br/>Objects bounce apart"]
    B -->|"0 < e < 1"| D["Inelastic<br/>KE partially lost<br/>Objects separate"]
    B -->|e = 0| E["Perfectly inelastic<br/>Max KE loss<br/>Objects stick together"]
    C --> F["Conserved: Momentum + KE"]
    D --> G["Conserved: Momentum only"]
    E --> G

Why This Works

Momentum conservation comes from Newton’s third law — during collision, forces are equal and opposite, so impulses cancel. KE conservation is an additional constraint that applies only when no energy is converted to other forms (heat, sound, deformation).

In real life, truly elastic collisions exist only at the molecular level (gas molecule collisions). All macroscopic collisions lose some KE. A bouncing ball on the floor is inelastic ( $e \approx 0.8$ ), and a ball of clay hitting a wall is perfectly inelastic ( $e = 0$ ).

Alternative Method

For JEE problems: in 1D elastic collisions between equal masses where one is initially at rest, the first object stops and the second moves with the first object’s velocity. This special case appears in problems about Newton’s cradle and billiard balls. Memorise it — it saves derivation time.

Common Mistake

Students assume kinetic energy is conserved in all collisions. This is the most fundamental error. KE is conserved ONLY in elastic collisions. In inelastic collisions, some KE is converted to heat, sound, or deformation energy. However, total energy (including heat, etc.) is always conserved — it is just the kinetic energy that is not. Momentum, on the other hand, is ALWAYS conserved regardless of collision type.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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