Elastic Collision Calculator for Momentum and Kinetic Energy

What this elastic collision calculator shows in one dimension

This elastic collision calculator models two blocks or carts moving along a straight track, then applies the textbook 1D elastic-collision equations to show how each mass and starting velocity changes the outcome. Because the interaction is perfectly elastic, the total momentum and total kinetic energy should stay the same before and after impact, and the canvas gives you a visual cue for which object rebounds, which one coasts through, and how strongly the mass ratio shapes the result. It is especially helpful when you want to compare a balanced pair of carts with a very heavy-and-very-light pair, since the same incoming speeds can lead to very different post-collision motion.

How to use the elastic collision calculator

1. Enter m₁, v₁, m₂, and v₂ (SI units: kg and m/s), then check that the two starting speeds point the way you expect before launching the animation.
2. Choose the time step Δt (smaller = smoother and usually more accurate; larger = faster but more likely to skip over a very brief impact, especially when the bodies are moving fast or just barely meet).
3. Press Play to start the elastic-collision animation; use Pause and Reset whenever you want to stop, inspect the current state, or restart from the initial spacing.
4. Use CSV to download the recorded time series of positions, velocities, and kinetic energies as a comma-separated file you can inspect in a spreadsheet or plotting tool.

Inputs, sign convention, and collision assumptions

• m₁, m₂: masses (must be positive and are treated as fixed values throughout the run).
• v₁, v₂: initial velocities. Positive is to the right; negative is to the left, so opposite signs mean the bodies are moving toward each other from opposite directions.
• v′₁, v′₂: velocities immediately after the collision, shown in the summary and used by the canvas once contact occurs.
• Δt: simulation time step used by the numerical update of positions and by the collision timing estimate when the blocks meet inside a frame.

This elastic-collision model assumes two rigid bodies moving on a frictionless line with no outside pushes, so the only changes in motion come from the impact itself. The calculator does not try to model rotation, bending, sound, or heat loss; instead, it focuses on the signed velocity change that comes from a perfectly elastic rebound. That makes it a good fit for textbook momentum problems, quick classroom demos, and any situation where you want the straight-line case without extra complications.

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

Governing formulas for this elastic collision calculator

Conservation of momentum in this elastic-collision calculator:

Formula: m_1 v_1 + m_2 v_2 = m_1 v_1^′ + m_2 v_2^′

Conservation of kinetic energy in the same elastic-collision setup:

Formula: 1 / 2 m_1 v_1^2 + 1 / 2 m_2 v_2^2 = 1 / 2 m_1 v_1^′^2 + 1 / 2 m_2 v_2^′^2

Solving those two equations yields the closed-form post-collision velocities that the animation uses:

• 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 elastic-collision results

• If m₁ = m₂, the calculator shows the two blocks exchange velocities exactly: v′₁ = v₂ and v′₂ = v₁.
• If m₁ ≫ m₂, block 1 keeps most of its original motion while block 2 can rebound with a much larger change in speed, which is the classic heavy-cart/light-cart behavior and one reason the mass ratio matters so much.
• If the initial velocities point away from each other, the blocks may never reach one another, so the calculator can show no impact at all depending on the starting gap and directions.

For each block, kinetic energy is still computed as KEᵢ = ½·mᵢ·vᵢ². In a perfectly elastic collision, the total momentum and total kinetic energy should match before and after impact, aside from tiny drift caused by the finite time step and discrete collision handling. If you notice a larger drift, reduce Δt and check that the bodies are actually moving toward each other. The readout is most informative when you compare the pre-impact and post-impact values rather than looking at either block in isolation, because the pair as a whole is what obeys the conservation laws.

Worked example: equal masses swapping speeds

Using the default values shown on this elastic-collision calculator: m₁ = 1 kg, v₁ = 2 m/s, m₂ = 1 kg, v₂ = −1 m/s.

Because the masses are equal, the two blocks 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. This is a useful sanity check because equal masses make the algebra simple enough to verify by eye, and the swap of velocities is exactly what the formula predicts when the two carts have the same mass.

Quick comparison: elastic collision calculator vs. inelastic collisions

This elastic collision calculator is specifically built for the perfectly elastic row of the table, but comparing it with the inelastic cases makes the role of energy conservation easier to see. In an inelastic collision, momentum still survives if no outside forces act, yet the final speeds are smaller because some of the initial kinetic energy ends up elsewhere. That contrast is one of the fastest ways to explain why elastic-collision problems are so useful in physics classes.

Numerical scheme for the elastic-collision animation (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 calculator estimates the collision moment, updates the velocities to v′₁ and v′₂, and then finishes the rest of the step with those new values. Smaller Δt usually reduces visible jitter and improves the energy bookkeeping, but it also increases the number of simulation steps. In practical use, the right step size is the one that keeps the impact visible without making the motion so coarse that you miss the moment when the bodies touch. If the setup is slow and simple, a larger step may be fine; if the carts approach each other quickly, a smaller step is worth the extra work.

Elastic-collision assumptions & limitations

• 1D only: the calculator does not model angles or multi-dimensional rebounds, so it is best thought of as a straight-line teaching tool.
• No external forces: friction, air drag, and slopes are ignored, which means the line of motion stays idealized from start to finish.
• No rotation or spin: only translational motion is represented, so the result is about speed and direction rather than tumbling or torque.
• Instantaneous collision: real contact time and deformation are not simulated, so the impact is treated as a clean event rather than a messy compression phase.
• Discrete time step: very large steps can miss a collision or create tiny conservation errors, so the animation works best when Δt is chosen with the motion speed in mind.
• Edge cases: extremely large speeds or masses may make the canvas harder to read; if the bodies never meet, no collision occurs and the calculator will simply continue the straight-line motion.