Elastic Collision

Forces
principle

Also known as: perfectly elastic collision

Grade 9-12

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A collision in which both the total momentum and the total kinetic energy of the system are fully conserved after impact. Elastic collisions model atomic and subatomic particle interactions and are the basis for understanding gas behaviour in the kinetic theory.

Definition

A collision in which both the total momentum and the total kinetic energy of the system are fully conserved after impact.

💡 Intuition

Billiard balls bouncing off each other: the total energy stays the same, nothing is lost to heat or deformation.

🎯 Core Idea

In an elastic collision, kinetic energy is fully recovered after impact.

Example

A steel ball bearing bouncing off another of equal mass — the first stops, the second moves at the same speed.

Formula

p_i = p_f \text{ and } KE_i = KE_f

Notation

m_1, m_2 are the masses, v_{1i}, v_{2i} are initial velocities, v_{1f}, v_{2f} are final velocities, p is momentum in kg·m/s, and KE is kinetic energy in joules.

🌟 Why It Matters

Elastic collisions model atomic and subatomic particle interactions and are the basis for understanding gas behaviour in the kinetic theory. They also approximate collisions between hard objects like billiard balls and Newton's cradle.

💭 Hint When Stuck

When solving an elastic collision problem, write two equations: one for conservation of momentum (m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}) and one for conservation of kinetic energy. Then solve the two equations simultaneously for the two unknown final velocities. A useful shortcut: the relative speed of approach equals the relative speed of separation.

Formal View

In an elastic collision: \sum m_i v_{i,\text{before}} = \sum m_i v_{i,\text{after}} and \sum \frac{1}{2}m_i v_{i,\text{before}}^2 = \sum \frac{1}{2}m_i v_{i,\text{after}}^2. Equivalently, the relative velocity reverses: v_{1i} - v_{2i} = -(v_{1f} - v_{2f}).

🚧 Common Stuck Point

Perfectly elastic collisions are rare in everyday life; most real collisions lose some energy.

⚠️ Common Mistakes

  • Assuming all bouncing collisions are perfectly elastic — most real collisions lose some kinetic energy to heat, sound, or deformation, even if objects bounce apart.
  • Using only conservation of momentum and neglecting the kinetic energy equation — elastic collisions require both conservation laws to solve for two unknowns.
  • Forgetting the shortcut: in elastic collisions, v_{1i} - v_{2i} = -(v_{1f} - v_{2f}), which can replace the energy equation and simplify algebra.

Frequently Asked Questions

What is Elastic Collision in Physics?

A collision in which both the total momentum and the total kinetic energy of the system are fully conserved after impact.

What is the Elastic Collision formula?

p_i = p_f \text{ and } KE_i = KE_f

When do you use Elastic Collision?

When solving an elastic collision problem, write two equations: one for conservation of momentum (m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}) and one for conservation of kinetic energy. Then solve the two equations simultaneously for the two unknown final velocities. A useful shortcut: the relative speed of approach equals the relative speed of separation.

How Elastic Collision Connects to Other Ideas

To understand elastic collision, you should first be comfortable with conservation of momentum and kinetic energy. Once you have a solid grasp of elastic collision, you can move on to inelastic collision.

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