Elastic Collision Final Velocities (1D, Two Bodies)

Elastic Collision Final Velocities (1D, Two Bodies)

An elastic collision conserves both momentum and kinetic energy — no energy is lost to heat, sound, or deformation, an idealization approached closely by hard objects like billiard balls or steel spheres in a Newton's cradle. Solving the two conservation equations simultaneously for two bodies yields closed-form expressions for both final velocities directly from the initial conditions.This is the reference case against which real, partially inelastic collisions (car crashes, most everyday impacts) are compared — the "coefficient of restitution" used in more general collision models is defined relative to exactly this elastic limit.

For a 1D elastic collision, the final velocities are v_1f = ((m_1−m_2)/(m_1+m_2))v_1i + (2m_2/(m_1+m_2))v_2i and v_2f = (2m_1/(m_1+m_2))v_1i + ((m_2−m_1)/(m_1+m_2))v_2i. where m_1e and m_2e are the two masses, v_1i and v_2i are their velocities before the collision, and v_1f and v_2f are their velocities after the collision.

Apply the standard elastic-collision formula for body 1's final velocity, which blends both initial velocities weighted by the relative masses.

Apply the matching formula for body 2's final velocity — together these two results automatically satisfy both momentum and kinetic energy conservation.

Results

Body 1 (the lighter, faster-approaching mass) rebounds with a reduced or reversed velocity while body 2 picks up speed in the original direction of body 1 — the expected outcome for a lighter object striking a heavier one nearly head-on. As a check, the total momentum m_1e·v_1f + m_2e·v_2f should equal the original total momentum m_1e·v_1i + m_2e·v_2i, and the total kinetic energy should also be unchanged since the collision is elastic.

Assumptions & limitations

• Perfectly elastic collision (kinetic energy conserved, no deformation).
• One-dimensional, head-on collision only.
• No external forces (friction, gravity component) during the collision.

Related calculations

• Projectile Motion Range
• Kinetic Energy
• Gravitational Potential Energy
• Linear Momentum