1D CollisionsJEE Main

Elastic, inelastic and restitution — interactive Physics simulation for IIT-JEE.

Concept

In any collision momentum is always conserved. Kinetic energy is conserved only in a perfectly elastic collision; otherwise some is lost to heat and sound. The coefficient of restitution $e$ measures how bouncy the collision is.

Key formula

e = \frac{v_2 - v_1}{u_1 - u_2}, \quad v_1 = \frac{(m_1-em_2)u_1 + (1+e)m_2u_2}{m_1+m_2}

Derivation

Conservation of momentum gives $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ . The restitution definition $e = \dfrac{\text{separation speed}}{\text{approach speed}}$ supplies the second equation. Solving the pair yields the final velocities above.

For $e=1$ (elastic) kinetic energy is conserved; for $e=0$ the bodies move together. Equal masses in an elastic head-on collision simply exchange velocities.

Scenarios to explore

1D Collisions — Elastic vs inelastic, momentum & restitution.

Real-world applications

Newton's cradle and billiards.
Vehicle crash analysis and safety design.
Particle scattering experiments.

JEE exam tips

Equal masses, elastic, head-on ⇒ velocities swap.
A perfectly inelastic collision loses the maximum KE consistent with momentum conservation.

Common mistakes

Assuming kinetic energy is conserved in every collision — only elastic ones.
Sign errors on velocities (pick one positive direction and stick to it).
Forgetting momentum is conserved even when energy is not.

Exam traps to avoid

$e$ depends only on the materials, not the masses or speeds.
Even in a 'totally inelastic' collision, momentum is still fully conserved.