1D CollisionsJEE Main

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

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.
• Equal masses, elastic: they swap velocities — A head-on elastic collision between equal masses hands over the whole motion.
• Perfectly inelastic: they stick — e = 0: the bodies leave together, losing the most kinetic energy that momentum allows.
• Heavy hits light — A 5 kg body hits a 0.5 kg one elastically: the light one flies off at nearly twice the speed.
• Partly elastic (e = 0.5) — Real collisions sit between the two extremes — watch the kinetic-energy bar.

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.