Conservation of Momentum (Two Bodies)

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

Worked example: 2 kg at 5 m/s into 3 kg at rest → v2 = 2.667 m/s — press Try an example to run it live, then adjust anything.

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Conservation of Momentum (Two Bodies) explained

With no external force acting, the total momentum of two colliding bodies before the impact equals the total after — always, whether they bounce, stick, or shatter. Kinetic energy is not so obliging; it survives only in a perfectly elastic collision. The principle was nailed down in 1668 when the Royal Society set the collision problem as a challenge and John Wallis, Christopher Wren and Christiaan Huygens independently sent in the answer.

Signs are everything in one dimension: pick a positive direction and stick with it, so a body moving the other way carries a negative velocity. A 2 kg cart at 5 m/s striking a stationary 3 kg cart and slowing to 1 m/s leaves the second cart at (10 + 0 − 2) ⁄ 3 ≈ 2.67 m/s. Because momentum is conserved in every collision, crash investigators use exactly this equation with skid-mark evidence to back out pre-impact speeds, and it works equally well for recoil: the rifle and the bullet start with zero total momentum and must end with zero.

Conservation of Momentum (Two Bodies) formula

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

Where

$m_1$ = Mass 1 (kg)
$u_1$ = Initial velocity 1 (m/s)
$m_2$ = Mass 2 (kg)
$u_2$ = Initial velocity 2 (m/s)
$v_1$ = Final velocity 1 (m/s)
$v_2$ = Final velocity 2 (m/s)

Missing one of these? Work it out first, then come back

Mass 1 — Newton's Second Law, Kinetic Energy
Initial velocity 1 — Final Velocity (Uniform Acceleration), Displacement (Uniform Acceleration)
Mass 2 — Newton's Second Law, Kinetic Energy
Initial velocity 2 — Final Velocity (Uniform Acceleration), Displacement (Uniform Acceleration)
Final velocity 1 — Final Velocity (Uniform Acceleration), Velocity-Displacement Relation (v² = v₀² + 2ad)
Final velocity 2 — Final Velocity (Uniform Acceleration), Velocity-Displacement Relation (v² = v₀² + 2ad)

Part of this formula set

Momentum, impulse and collisions

7 formulasMomentum and impulse, the conservation law that links two bodies, and the elastic, inelastic and restitution results for what happens after they meet.