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1D Momentum Collision & Impulse Calculator

Linear momentum (p = m·v) is strictly conserved in all isolated physical systems during collisions.

Mass of the first colliding body in kilograms.

Pre-collision velocity of mass 1 (positive = right, negative = left).

Mass of the second colliding body in kilograms.

Pre-collision velocity of mass 2.

1.0 = Perfectly Elastic (no KE lost); 0.0 = Perfectly Inelastic (stick together).

Calculated Result
0.00 m/s, 3.00 m/s

Final Velocities (v₁ & v₂)

Initial Momentum

6.00 kg·m/s

Final Kinetic Energy

9.00 J

Energy Dissipated

0.00 J (0.0%)

Impulse Transferred

-6.00 N·s

Calculation Breakdown

  1. Initial Total System Momentum6.000 kg·m/spi=m1u1+m2u2p_i = m_1 u_1 + m_2 u_2
  2. Final Velocities Post-Collisionv₁ = 0.00 m/s, v₂ = 3.00 m/sv1=pi−m2e(u1−u2)m1+m2,v2=pi+m1e(u1−u2)m1+m2v_1 = \frac{p_i - m_2 e (u_1 - u_2)}{m_1 + m_2}, \quad v_2 = \frac{p_i + m_1 e (u_1 - u_2)}{m_1 + m_2}
  3. System Kinetic Energy BalanceInitial: 9.00 J → Final: 9.00 J (Dissipated: 0.00 J / 0.0%)ΔKE=KEi−KEf\Delta KE = KE_i - KE_f
  4. Impulse Transferred on Body 1-6.00 N·sJ1=m1(v1−u1)J_1 = m_1 (v_1 - u_1)

Kinetic Energy Dissipation vs Restitution Coefficient

Interactive visualization based on your current inputs

Energy Lost (%)
0.013253850e = 0.0 (Inelastic)e = 0.25e = 0.50e = 0.75e = 1.0 (Elastic)Restitution (e)KE Lost (%)

What Is the 1D Momentum Collision & Impulse Calculator?

Collisions represent brief physical interactions between moving bodies where internal contact forces dwarf external forces.

Analyzing momentum and restitution reveals velocity changes, energy dissipation, and safety impact forces.

How Does the 1D Momentum Collision & Impulse Calculator Work?

The law of conservation of momentum mandates that total system momentum before impact equals total momentum after impact.

The coefficient of restitution (e) defines the ratio of relative separation speed to relative approach speed.

Solving these two linear equations simultaneously gives the final post-impact velocities.

1D Momentum Collision & Impulse Calculator Formula & Variables

The core mathematical equation utilized by this calculator is expressed as:

v1=m1u1+m2u2−m2e(u1−u2)m1+m2,v2=m1u1+m2u2+m1e(u1−u2)m1+m2v_1 = \frac{m_1 u_1 + m_2 u_2 - m_2 e(u_1 - u_2)}{m_1 + m_2}, \quad v_2 = \frac{m_1 u_1 + m_2 u_2 + m_1 e(u_1 - u_2)}{m_1 + m_2}

Conservation of momentum coupled with Poisson’s coefficient of restitution determines post-collision rebound velocities.

How to Use the 1D Momentum Collision & Impulse Calculator

  1. Enter the masses and initial velocities for both bodies.
  2. Set the coefficient of restitution (1.0 for steel/glass billiards, 0.5–0.8 for sports balls, 0.0 for clay or car crashes).

Step-by-Step Example Calculation

Equal Mass Billiard Ball Head-On Impact

Input Values:

mass1:2
velocity1:3
mass2:2
velocity2:0
restitution:1
Worked Steps: With equal masses and e = 1, Mass 1 comes to a dead stop (v₁ = 0 m/s) and Mass 2 continues forward at the full 3 m/s with zero energy loss.

Understanding Your Result

Final Velocities (v₁ & v₂): Post-collision speeds and directions.

Total Momentum: Unchanged system momentum (conserved).

Kinetic Energy Loss: Thermal and sound dissipation caused by inelastic deformation.

Factors That Affect the Result

  • Mass Ratio: A light object striking a massive stationary wall rebounds at nearly its original speed reversed.
  • Restitution: Controls the fraction of kinetic energy preserved.

When Should You Use This Calculator?

  • Crash analysis, sports physics, ballistic pendulum testing, and aerospace docking simulations.

Assumptions & Limitations

  • Assumes collinear one-dimensional motion along a straight line.
  • Assumes zero friction with the floor during the impact interval.

Frequently Asked Questions

Calculation Accuracy & Reference Note

Deterministic closed-form solution to Newton-Poisson collision mechanics.

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