Elastic Collision Simulator

What this simulator calculates

This page models a one‑dimensional, perfectly elastic collision between two blocks (or carts) moving along a straight line. You enter each block’s mass and initial velocity, then the simulator computes the post‑collision velocities using conservation laws and animates the motion. It also tracks kinetic energy over time and can export the time series to CSV.

How to use

1. Enter m₁, v₁, m₂, v₂ (SI units: kg and m/s).
2. Choose the time step Δt (smaller = smoother/more accurate; larger = faster but less precise).
3. Press Play to start the animation; use Pause and Reset as needed.
4. Use CSV to download the recorded time series (positions, velocities, and energies if included by the implementation).

Variables, sign convention, and assumptions

• m₁, m₂: masses (must be positive).
• v₁, v₂: initial velocities. Positive is to the right; negative is to the left.
• v′₁, v′₂: velocities immediately after the collision.
• Δt: simulation time step used by the numerical update of positions.

The model assumes two rigid bodies moving on a frictionless line. The collision is instantaneous and perfectly elastic, meaning:

• Total linear momentum is conserved.
• Total kinetic energy is conserved.

Governing formulas (perfectly elastic, 1D)

Conservation of momentum:

Conservation of kinetic energy:

Solving these yields the standard closed‑form results:

• Block 1 after collision:
  v′₁ = ((m₁ − m₂)/(m₁ + m₂))·v₁ + (2m₂/(m₁ + m₂))·v₂
• Block 2 after collision:
  v′₂ = (2m₁/(m₁ + m₂))·v₁ + ((m₂ − m₁)/(m₁ + m₂))·v₂

Interpreting the results

• If m₁ = m₂, the blocks exchange velocities: v′₁ = v₂ and v′₂ = v₁.
• If m₁ ≫ m₂, block 1 changes speed only slightly, while block 2 can rebound with a much larger change (similar to a heavy cart hitting a light cart).
• If the objects are not actually approaching (for example, they’re moving apart), an ideal collision may never occur; the animation may show no impact depending on initial spacing and direction.

Kinetic energy for each block is:

In a perfectly elastic collision, total kinetic energy and total momentum stay constant. In the simulator, you may see tiny drift due to finite time steps and discrete collision handling.

Worked example

Using the default values shown on the page: m₁ = 1 kg, v₁ = 2 m/s, m₂ = 1 kg, v₂ = −1 m/s.

Because the masses are equal, they exchange velocities:

• v′₁ = v₂ = −1 m/s
• v′₂ = v₁ = 2 m/s

Check momentum: initial p = 1·2 + 1·(−1) = 1; final p = 1·(−1) + 1·2 = 1. Check energy: initial KE = ½·1·2² + ½·1·(−1)² = 2 + 0.5 = 2.5 J; final KE = ½·1·(−1)² + ½·1·2² = 0.5 + 2 = 2.5 J.

Quick comparison (elastic vs inelastic)

Numerical scheme (why Δt matters)

The animation advances time in increments of Δt. Between collisions, each block moves at constant velocity, so positions are updated using a simple step:

When the blocks overlap or their edges cross between steps, the simulator triggers a collision response and replaces the velocities with v′₁ and v′₂. Smaller Δt usually reduces visible jitter and improves conservation behavior in the recorded data, at the cost of more steps.

Assumptions & limitations

• 1D only: no angles, no 2D/3D motion.
• No external forces: friction, air drag, and slopes are ignored.
• No rotation or spin: only translational motion is modeled.
• Instantaneous collision: real contact time and deformation are not simulated.
• Discrete time step: small numerical errors can accumulate; results depend slightly on Δt and collision handling.
• Edge cases: extremely large speeds/masses may cause visualization scaling issues; if objects never meet, no collision occurs.