Elastic Collision Calculator

Solve 1D perfectly elastic collisions for initial or final velocities using conservation laws.

Configuration

Choose what you want to solve for, then enter the known masses and velocities.

Known Values

Result

Practical Guide

Elastic Collision Calculator: How to Solve Perfectly Elastic 1D Impacts

This guide shows you how to use the Elastic Collision Calculator correctly, interpret the outputs, and understand the conservation-law assumptions behind perfectly elastic collisions. You’ll also find worked examples, common variations, and fast sanity checks for homework, lab work, and real engineering estimates.

6–8 min read Updated 2025 Physics / Dynamics

Quick Start

The calculator handles 1D perfectly elastic collisions (head-on impacts along a single line). Sign matters: choose a positive direction and keep it consistent. Use negative velocities for motion opposite your chosen direction.

  1. 1 Pick a sign convention. For example, “to the right is positive.”
  2. 2 In Solve For, choose what you need: Final velocities (v₁, v₂) for typical problems, or Initial velocity of object 1/2 (u₁ or u₂) to “rewind” an experiment.
  3. 3 Enter m₁ and m₂. Use any mass units—you just need them consistent.
  4. 4 For Final velocities, enter the known u₁ and u₂. The v-rows will auto-hide because they’re being solved.
  5. 5 For Initial velocity, enter the known v₁ and v₂. The u-rows will auto-hide because they’re being solved.
  6. 6 Set Output Velocity Units (m/s, ft/s, km/h, mph). The result row and quick stats update together.
  7. 7 Read the main result, then glance at Quick Stats to confirm momentum and kinetic energy match.

Tip: If one object starts “at rest,” enter its velocity as 0, not blank. Leaving a required field empty prevents calculation.

Common mistake: Mixing directions. If u₁ is +4 m/s and u₂ is −2 m/s, the objects are moving toward each other. If both are positive, they’re moving the same way.

Under the hood, the calculator uses conservation of momentum and kinetic energy: \[ m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \] \[ \tfrac12 m_1u_1^2 + \tfrac12 m_2u_2^2 = \tfrac12 m_1v_1^2 + \tfrac12 m_2v_2^2 \] These apply only when the collision is perfectly elastic, meaning no energy is lost to heat, sound, or deformation.

Choosing Your Method

There are a few standard ways to analyze elastic collisions. They’re equivalent, but some are faster depending on the data you have. The calculator follows Method A by default.

Method A — Closed-Form 1D Elastic Equations

Best when you know masses and initial velocities and want final velocities immediately.

  • Fastest for homework and design checks.
  • No iteration; direct algebraic solution.
  • Works for any mass ratio.
  • Only valid for 1D, perfectly elastic impacts.
  • Harder to generalize to partial elasticity.
\( v_1=\frac{m_1-m_2}{m_1+m_2}u_1+\frac{2m_2}{m_1+m_2}u_2 \), \( v_2=\frac{2m_1}{m_1+m_2}u_1+\frac{m_2-m_1}{m_1+m_2}u_2 \)

Method B — Relative Velocity / Coefficient of Restitution

Useful if you’re comparing elastic vs inelastic cases or measuring “bounciness.”

  • Highlights the physical meaning of elasticity.
  • Extends naturally to \(e<1\) (inelastic collisions).
  • Still needs momentum conservation to close the system.
  • More algebra if you want final velocities.
\(e=\dfrac{v_2-v_1}{u_1-u_2}\), and for perfectly elastic \(e=1\Rightarrow v_2-v_1=u_1-u_2\).

Method C — Center-of-Mass (COM) Frame

Great for intuition: in the COM frame, velocities reverse in an elastic collision.

  • Conceptually clean; reduces chance of sign errors.
  • Good for explaining Newton’s cradle behavior.
  • Extra step to transform to/from COM frame.
  • Not as quick for plug-and-chug problems.
\(v’_{1}=-u’_{1}\), \(v’_{2}=-u’_{2}\) in the COM frame, then transform back.

If you’re strictly in 1D with \(e=1\), Method A is the most efficient. If your lab data shows \(e\neq1\), this calculator is not the right tool—use an inelastic collision model instead.

What Moves the Number the Most

Elastic collision outcomes are sensitive to a few dominant levers. Understanding these helps you sanity-check results quickly.

Mass ratio \(m_1/m_2\)

The bigger the ratio, the more object 1 “plows through” object 2. If \(m_1\gg m_2\), object 1 barely slows, and object 2 shoots off near \(2u_1-u_2\).

Initial relative speed \(u_1-u_2\)

Elasticity preserves the magnitude of relative speed, just flips direction: \(v_2-v_1 = u_1-u_2\). So if the approach speed is large, the separation speed will be large.

Direction / sign convention

A sign error changes the physics. Always encode “toward each other” with opposite signs. If both velocities share the same sign, they’re moving in the same direction.

Target at rest vs moving

When \(u_2=0\), the classic results appear. For equal masses, velocities swap. When the target moves, you’re effectively shifting to a moving reference frame.

Equal-mass limit

If \(m_1=m_2\), then \(v_1=u_2\) and \(v_2=u_1\). That’s why Newton’s cradle transfers speed cleanly across identical balls.

Dimensional consistency

Units don’t change the math, but mixing them does. Use the unit selectors so all velocities are internally normalized to m/s before solving.

Worked Examples

Example 1 — Steel cart hits a lighter cart (solve finals)

  • m₁: 2.0 kg
  • m₂: 0.5 kg
  • u₁: +3.0 m/s
  • u₂: 0.0 m/s (at rest)
  • Goal: Find \(v_1\) and \(v_2\)

Use the closed-form elastic solutions: \[ v_1=\frac{m_1-m_2}{m_1+m_2}u_1+\frac{2m_2}{m_1+m_2}u_2 \quad,\quad v_2=\frac{2m_1}{m_1+m_2}u_1+\frac{m_2-m_1}{m_1+m_2}u_2 \]

1
Compute denominator: \(m_1+m_2=2.0+0.5=2.5\ \text{kg}\).
2
\(v_1=\frac{2.0-0.5}{2.5}(3.0)+\frac{2(0.5)}{2.5}(0)\) \(=\frac{1.5}{2.5}\cdot3.0=1.8\ \text{m/s}\).
3
\(v_2=\frac{2(2.0)}{2.5}(3.0)+\frac{0.5-2.0}{2.5}(0)\) \(=\frac{4.0}{2.5}\cdot3.0=4.8\ \text{m/s}\).
4
Sanity check: lighter cart leaves faster than the heavy cart, which makes physical sense.

Entering these numbers into the calculator yields the same results and shows both conservation checks near zero error.

Example 2 — Equal masses, opposite directions (solve finals)

  • m₁: 1.2 kg
  • m₂: 1.2 kg
  • u₁: +2.5 m/s
  • u₂: −1.0 m/s
  • Goal: Find \(v_1\) and \(v_2\)

For equal masses, velocities swap: \[ v_1=u_2,\qquad v_2=u_1 \] So: \[ v_1=-1.0\ \text{m/s}\quad,\quad v_2=+2.5\ \text{m/s}. \]

1
Confirm equality: \(m_1=m_2\Rightarrow\) swap rule applies.
2
Apply swap: \(v_1=u_2=-1.0\ \text{m/s}\).
3
\(v_2=u_1=+2.5\ \text{m/s}\).
4
Relative speed preservation: \(u_1-u_2=2.5-(-1.0)=3.5\), \(v_2-v_1=2.5-(-1.0)=3.5\).

This case is a great way to debug sign conventions. If your calculator doesn’t return swapped velocities, the inputs are probably in the wrong direction.

Example 3 — Rewind a lab (solve for initial velocity)

  • m₁: 0.20 kg
  • m₂: 0.30 kg
  • Measured v₁: −0.50 m/s
  • Measured v₂: +0.80 m/s
  • Goal: Recover \(u_1\) and \(u_2\) (perfectly elastic assumption)

Because elastic collisions are time-reversible, the same closed-form equations apply with v’s as inputs: \[ u_1=\frac{m_1-m_2}{m_1+m_2}v_1+\frac{2m_2}{m_1+m_2}v_2 \quad,\quad u_2=\frac{2m_1}{m_1+m_2}v_1+\frac{m_2-m_1}{m_1+m_2}v_2 \]

1
Denominator: \(m_1+m_2=0.50\ \text{kg}\).
2
\(u_1=\frac{0.20-0.30}{0.50}(-0.50)+\frac{2(0.30)}{0.50}(0.80)\) \(=(-0.20)(-0.50)+(1.20)(0.80)=0.10+0.96=1.06\ \text{m/s}\).
3
\(u_2=\frac{2(0.20)}{0.50}(-0.50)+\frac{0.30-0.20}{0.50}(0.80)\) \(=(0.80)(-0.50)+(0.20)(0.80)=-0.40+0.16=-0.24\ \text{m/s}\).

If your measured momentum or energy checks are far from zero, your collision was not perfectly elastic, so these recovered u-values are only an approximation.

Common Layouts & Variations

Even in 1D, collisions show repeatable patterns. The table below summarizes typical configurations you’ll see in dynamics problems and lab setups.

ConfigurationTypical InputsExpected BehaviorPros / Where UsedCons / Pitfalls
Target at rest\(u_2=0\)Object 2 departs faster as \(m_2\) decreases.Most common textbook and safety-buffer design cases.Easy to forget sign if “rest” isn’t on same axis.
Equal masses\(m_1=m_2\)Velocities swap: \(v_1=u_2,\ v_2=u_1\).Newton’s cradle, identical carts/balls.Swap rule fails if collision isn’t perfectly elastic.
Heavy striker\(m_1\gg m_2\)\(v_1\approx u_1\), \(v_2\approx 2u_1-u_2\).Ball-bat impacts, recoil approximations.Assumption breaks if deformation dominates.
Heavy target (wall limit)\(m_2\gg m_1\)Object 1 rebounds: \(v_1\approx -u_1+2u_2\).Bouncing off fixed barriers.Wall is rarely perfectly elastic in reality.
Opposite directions\(u_1>0\), \(u_2<0\)Higher approach speed → higher separation speed.Vehicle crash idealizations, particle collisions.Most common source of sign mistakes.

Interpretation hint: In 1D elastic collisions, the relative speed is preserved: \(v_2-v_1=u_1-u_2\). If that doesn’t hold, revisit inputs or elasticity assumptions.

Specs, Logistics & Sanity Checks

Whether you’re using this for coursework or for a quick engineering approximation (e.g., impact testing, toy-model crash dynamics, or particle work), verify these items before trusting the output.

Assumptions to Confirm

  • Motion is effectively 1D (head-on along a line).
  • Collision is perfectly elastic (\(e=1\)).
  • Negligible external impulse during impact (short contact time).
  • Masses are constant and well-measured.

Data Collection Notes

  • Mark a clear axis and record direction.
  • Use consistent units; avoid mixing mph with m/s.
  • Average repeated trials to reduce noise.
  • Watch for rotational energy—this violates 1D translation-only models.

Sanity Checks After Solving

  • Momentum before ≈ momentum after.
  • Kinetic energy before ≈ kinetic energy after.
  • Equal masses should swap speeds.
  • Light target should leave faster than heavy striker.

If the conservation checks are noticeably off (more than a few percent), the collision likely isn’t perfectly elastic. Causes include deformation, friction, sound losses, or off-axis motion. In that case, use a coefficient of restitution model with \(0<e<1\) or measure the energy loss directly.

Frequently Asked Questions

What does “perfectly elastic” mean in this calculator?
Perfectly elastic means both linear momentum and kinetic energy are conserved during the collision. Mathematically, the coefficient of restitution is \(e=1\). Real collisions are usually slightly inelastic, so treat perfectly elastic as an ideal upper-bound case.
Can I use this for 2D or angled collisions?
Not directly. This tool is for 1D head-on impacts. In 2D you need to resolve velocities into components and handle vector momentum; energy conservation may still apply, but the algebra differs.
Why do u₁/u₂ rows disappear when solving for initial velocities?
When you pick “solve for initial velocity,” the calculator assumes you have final velocities \(v_1, v_2\) and rewinds the collision. The u-rows are hidden and disabled so unused fields never affect the computation.
What if my collision isn’t perfectly elastic?
Then kinetic energy is not conserved. Use an inelastic model with \(e=\dfrac{v_2-v_1}{u_1-u_2}\) where \(0<e<1\), or measure the energy loss. This calculator will over-predict rebound speeds if \(e<1\).
Do the mass units have to match?
Yes. You can choose kg, g, or lbm in the unit selectors, but m₁ and m₂ must be in consistent units so the ratio \(m_1/m_2\) is correct. The calculator converts everything internally to SI.
Why are my results “swapped” for equal masses?
That’s the expected elastic outcome. If \(m_1=m_2\), the objects exchange velocities: \(v_1=u_2\) and \(v_2=u_1\). Newton’s cradle works because of this property.
What’s the fastest way to check if I typed directions correctly?
Compute the relative speed before and after. For elastic collisions, \(v_2-v_1=u_1-u_2\). If the magnitudes don’t match, you likely flipped a sign or used the wrong direction for one object.
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