Enter the mass, initial velocity, and final velocity of object 1, and the mass and initial velocity of object 2 to calculate the final velocity of object 2 using conservation of linear momentum. This applies to a 1D collision/interaction where the net external impulse on the two-object system is negligible (elastic or inelastic). See also our elastic collision calculator.

Conservation of Momentum (1D)

Enter the masses and velocities below; the calculator finds Final Velocity 2 and shows the system momentum before and after. Use a sign convention (velocities in the opposite direction should be negative).

Conservation of Momentum Formula

The following formula is used for two objects interacting in one dimension when the net external impulse on the system is negligible (so total momentum is conserved).

\begin{aligned}
m_1 u_1 + m_2 u_2 &= m_1 v_1 + m_2 v_2 \\
v_2 &= \frac{m_1 u_1 + m_2 u_2 - m_1 v_1}{m_2}
\end{aligned}
  • Where m₁ is the mass of object 1
  • u₁ is the initial velocity of object 1
  • v₁ is the final velocity of object 1
  • m₂ is the mass of object 2
  • u₂ is the initial velocity of object 2
  • and v₂ is the final velocity of object 2

Choose a positive direction along the line of motion and use positive/negative velocities accordingly. Conservation of momentum states that the system’s total momentum before the interaction equals the system’s total momentum after the interaction (assuming negligible net external impulse).

Conservation of Momentum Definition

Conservation of momentum is a law of physics that says the total momentum of an isolated system remains constant. In other words, the system’s total momentum can only change if there is a net external impulse (e.g., due to outside forces) on the system.

How to calculate the conservation of momentum?

How to use conservation of momentum (1D) to solve for an unknown velocity.

  1. First, define a sign convention and list known values.

    Choose a positive direction along the line of motion, then write down m₁, m₂, and the known initial/final velocities (u₁, u₂, v₁, v₂).

  2. Next, write and rearrange the momentum equation.

    Use m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ and rearrange to solve for the unknown variable.

  3. Finally, calculate the unknown and (optionally) verify.

    Compute the missing value and check that total momentum before and after matches (within rounding).

FAQ

What is conservation of momentum?

Conservation of momentum is a law of physics that says the total momentum of an isolated system remains constant. In other words, the system’s total momentum can only change if there is a net external impulse (e.g., due to outside forces) on the system.

How does momentum change?

Total momentum of a system changes when there is a net external impulse (net external force over time). For a single object in classical mechanics, p = mv, so (at constant mass) its momentum changes when its velocity changes.