⚛ Physics LabMomentum and Collisions · v0.26 beta

DYNAMICS · CONSERVATION OF MOMENTUM

Momentum and Collisions

Simulate two bodies colliding in one dimension. Explore conservation of momentum , the impulse exchange, the coefficient of restitution and kinetic energy loss in inelastic collisions.

Physical Parameters

kg
kg
m
m
m
m
m/s
m/s
dimensionless
s

Sign Convention: the x axis points to the right. Positions x₁ and x₂ are the centres of the spheres: initially they must be separated by at least r₁ + r₂. Their surfaces touch when x₂ − x₁ = r₁ + r₂; overlapping is not allowed. Coefficient e ranges from 0 (perfectly inelastic) to 1 (elastic).

Examples: Equal masses · Unequal masses · Inelastic collision · Partial collision · Head-on collision · Catch-up collision · No collision

Two-Body Simulation

True Geometric Scale 1:1: sphere radii and distances between their centres use the same scale. The view fits the full motion without allowing spheres to overlap.

Body ABody BCollision Time

Before/After Collision Balance

Initial Momentum—
Final Momentum—
Initial Energy—
Final Energy—

Position A—
Position B—
Surface Clearance—
Velocity A—
Velocity B—
Total Momentum—
Total Kinetic Energy—
Dissipated Energy—
Impulse on A—
Impulse on B—
Momentum Residual—
Energy Balance Residual—
Collision Time—

The collision is instantaneous: impulse abruptly changes velocities at contact. A finite constant force or ordinary instantaneous acceleration is not assigned to the impact. If e = 0, both spheres continue at the same velocity while remaining tangent.

Synchronized Graphs

Vertical jumps in velocity, individual momentum, energy and impulse represent an instantaneous impact, not a continuous change over time. Click the graphs to select a time.

Selected time: 0.00 s

Position of Both Bodies x(t)

Velocity of Both Bodies v(t)

Momentum p(t), Including the Total

Kinetic Energy Before and After the Collision

Cumulative Impulses on Both Bodies

Values Table

t (s)Phasex A (m)x B (m)Surface clearance (m)v A (m/s)v B (m/s)p A (kg·m/s)p B (kg·m/s)total p (kg·m/s)total K (J)J A (N·s)J B (N·s)Q (J)

CSV in SI units, comma-separated with decimal points. Includes both the pre-impact and post-impact states at the same time.

Model Assumptions

Two rigid spheres with adjustable radii move along a single inertial axis, with no external forces, friction or gravity along the line of motion. Coordinates locate the centres, and contact occurs when their separation equals the sum of the radii.

The isolated system conserves momentum. The collision is instantaneous with a constant coefficient of restitution 0 ≤ e ≤ 1. Detailed deformation and repeated wall impacts are omitted. When e = 0 the spheres stay in contact without overlapping and move together. Masses and radii are independent (no fixed density is imposed).

Kinetic energy is conserved only for e = 1. Otherwise the positive difference is accounted for as energy transferred to nonmechanical forms (deformation, heat and sound), not as energy lost from the universe.

If A does not catch B within the chosen duration, the program reports “no collision expected” instead of artificially applying collision formulas.

CONSERVATION AND IMPULSE

Step-by-Step Calculations

General formulas, numerical substitutions and dimensional checks.

Each equation occupies a separate left-aligned line, with a MathML fallback if MathJax is unavailable. Rounding affects presentation only.

Other Labs: Work and Energy · Newton’s Laws and Forces · Physics Lab

by Andrea Minini