Physics: 1D Restitution Collision Calculator

What is Newtonian Collisions: Conservation of Momentum and Kinetic Bleed?

When two objects collide linearly on a 1D track, two rigid mathematical rules completely dictate the physical aftermath. The First Rule is the absolute Conservation of Momentum—Sir Isaac Newton's mandate that the total absolute push of the system before the crash must perfectly equal the push after the crash. The Second Rule is governed by the Coefficient of Restitution (e), which measures molecular deformation. Objects that perfectly bounce apart (e=1) conserve all kinetic energy, while objects that shatter or stick together (e<1) bleed massive amounts of kinetic energy into the surrounding environment as heat and sound.

Mathematical Foundation

Final Velocity (Object 1)

v_{1f} = \frac{m_1 v_{1i} + m_2 v_{2i} + m_2 e (v_{2i} - v_{1i})}{m_1 + m_2}

v_{1f}

= The final post-collision velocity vector of Object 1 (m/s). A negative result indicates the object was physically repulsed and is traveling backwards.

m_1, m_2

= The unalterable physical Mass of each object (kg). Total system mass strictly defines the inertia limits of the collision.

v_{1i}, v_{2i}

= Initial velocity vectors (m/s) before impact. Crucially, a head-on collision mathematically mandates that one vector MUST be negative (objects traveling toward each other).

e

= Coefficient of Restitution (Dimensionless ratio). Defines the 'bounciness' of the objects, securely clamping the physical outcome between e = 0.0 (Perfectly Inelastic / Mud) and e = 1.0 (Perfectly Elastic / Billiard Balls).

Laws & Principles

Conservation of Momentum is Inviolable: In a closed system, total momentum (Mass × Velocity) mathematically never changes, regardless of how violently the objects shatter or bounce. If a 10 kg bowling ball moving right at 5 m/s hits a stationary 10 kg ball, the system possesses exactly +50 Momentum Vectors before the crash. After the crash, regardless of the restitution coefficient, the combined momenta of the two balls MUST still equal exactly +50. If the calculator yields a momentum delta, the physics engine has mathematically failed.
The Perfectly Elastic Boundary (e = 1.0): This represents an idealized physics scenario where two indestructible objects (e.g., rigid billiards balls or subatomic particles) violently collide without any molecular thermal deformation. They instantly bounce and perfectly preserve 100.0% of the entire system's kinetic energy. Zero Joules are lost to heat or sound.
The Perfectly Inelastic Trap (e = 0.0): This occurs when two heavily deformable objects (e.g., sticky wet clay or crashing automobiles) violently hit. The Restitution factor drops blindly to 0. They physically fuse into one heavy, slow, coupled mass. Because momentum is conserved but restitution is zero, they massively bleed kinetic energy, permanently mutating shapes and converting physical velocity into ambient thermal heat deformation.

Step-by-Step Example Walkthrough

" A heavy 2.0 kg ball travelling Right (+5.0 m/s) destroys a smaller 1.5 kg ball traveling Left (-3.0 m/s) head-on. The material is hard steel, executing a perfectly elastic collision (e = 1.0). "

1. Calculate the foundational momentum of the board: p_initial = (m1 * v1i) + (m2 * v2i) = (2 * 5) + (1.5 * -3) = 10 - 4.5 = +5.5 kg·m/s.
2. Analyze the denominator limitation: The total mass sharing the kinetic impact is rigidly 3.5 kg (2 + 1.5).
3. Process the explicit restitution bounds: e = (v2f - v1f) / (v1i - v2i) = 1.0.
4. Execute the velocity resolution matrix for Object 1: v1f = [(2 * 5) + (1.5 * -3) + 1.5 * 1.0 * (-3 - 5)] / 3.5.
5. Resolve the numerator: [10 - 4.5 + (-12)] = -6.5 m/s. Divide by 3.5 mass = -1.86 m/s.
6. Execute parallel resolution for Object 2: v2f = +6.14 m/s.

Final Result: The heavy 2.0 kg ball abruptly reverses direction to -1.86 m/s (violently pushed backwards). The lighter 1.5 kg ball absorbs the massive impact transfer and is utterly blasted forward to the Right at +6.14 m/s. Total momentum remains strictly locked at +5.5 kg·m/s.

Quick Answer: How do you calculate final velocity after a collision?

To calculate final collision velocities, you must know the Mass and Initial Velocity of both objects, plus their Coefficient of Restitution (e). The mathematical engine fuses Newton's Conservation of Momentum (Total Initial p = Total Final p) with kinematic restitution bounds to isolate the exact final vector velocity of both physical fragments. If the objects stick together completely (e = 0), they simply merge their momentums and share a single uniformly decayed velocity trajectory.

Collision Type Parameter Matrix

A standard reference for recognizing the structural damage limitations defined by Restitution (e).

Collision Type	Restitution (e)	Momentum	Kinetic Energy	Real World Limit
Perfectly Elastic	e = 1.0	100% Conserved	100% Conserved	Subatomic particles, Ideal Billiards
Partially Elastic	0.0 < e < 1.0	100% Conserved	Lost to Heat/Sound	Basketballs, Tennis Balls (e ≈ 0.75)
Perfectly Inelastic	e = 0.0	100% Conserved	Maximum Bleed	Wet Clay, Auto Crashes (Objects fuse)
Super-Elastic / Explosive	e > 1.0	Violated	Violated (Created)	Explosive Detonations (Requires added chemical energy)

Pro Tips & Common Calculation Mistakes

Do This

✓Strictly enforce directional vector signs (+/-). Velocity is a vector—it fundamentally posesses an absolute direction. If two objects are crashing head-on, their velocity inputs MUST have opposite signs (e.g., Object 1 is +5 m/s, Object 2 is -3 m/s). Inputting both as positive mathematically simulates Object 1 chasing Object 2 from behind.
✓Verify the system momentum checksum. No matter how complex the inelastic crash damage is, the `Total Momentum Before` must perfectly, to the exact decimal, match the `Total Momentum After`. If these values drift, you have suffered a mathematical sign error inside the Restitution division.

Avoid This

✗Don't confuse Momentum with Kinetic Energy. They scale differently. Momentum scales linearly ($p = mv$), but Kinetic Energy scales quadratically ($KE = 0.5 \cdot m \cdot v^2$). Therefore, doubling the speed of an incoming car only doubles its momentum push, but completely quadruples its explosive destructive kinetic energy.
✗Don't pass negative Restitution bounds. The Coefficient of Restitution cannot physically pierce below exactly 0.0 or above exactly 1.0 (unless executing a controlled explosive blast). An input of e = 1.5 implies the collision spontaneously magically created kinetic energy out of nowhere, severely violating Newtonian bounds.

Frequently Asked Questions

What does it mean if a Post-Collision Velocity is negative?

Because velocity is tracked as a physical vector, the positive/negative sign strictly defines global direction. If Object 1 initially travels Right at +5 m/s, and its final calculated velocity is -2 m/s, it was physically overpowered by the crash and is now violently travelling backward to the Left. The negative sign is a crucial architectural output, not a math error.

Where does the lost Kinetic Energy go in an Inelastic (e = 0) collision?

The First Law of Thermodynamics mandates that energy cannot be utterly destroyed. In highly inelastic crashes (like car wrecks or dropping raw dough), the massive volume of "lost" kinetic energy is simply converted. It bleeds away structurally—tearing metal, shattering molecular bonds, emitting deafening soundwaves, and violently raising the localized ambient temperature (heat friction).

Why don't we need to know the time duration of the crash?

Basic collision algorithms map identical "Before" and "After" momentum snapshots, deliberately bypassing the chaotic micro-seconds of the active impact. Unless you explicitly physically need to calculate the exact Force generated (Impulse = Force × Time), the strict conservation laws allow you to predict the outcome vectors without knowing if the crash took 0.001 seconds (hard steel) or 2.0 seconds (crumple zones).

Physics: 1D Restitution Collision Calculator

Physics: 1D Restitution Collision

Target Entity 1

Target Entity 2

Final Vel 1

Final Vel 2