Collision of Particles Simulation

Understanding Coulomb Repulsion Through Motion

This interactive simulation explores what happens when two positively charged nuclei move toward one another along a straight line. Because both nuclei carry positive charge, the electric force between them is repulsive. As their separation decreases, that repulsion becomes stronger, slowing their approach until they reach a minimum separation and move apart again.

Why the particles do not meet

The nuclei begin with kinetic energy, but approaching each other increases their electric potential energy. Their initial motion is gradually converted into potential energy by the Coulomb interaction. At the point of closest approach, the relative velocity is zero: for one instant, the distance between the nuclei stops decreasing. They then accelerate away from each other as electric potential energy is converted back into kinetic energy.

Momentum and the centre of mass

The two nuclei exert equal and opposite forces on each other. These are internal forces, so the total momentum of the two-particle system remains constant. The centre of mass therefore moves at a constant velocity even though the velocity of each individual nucleus changes. The purple marker in the spatial simulation makes this collective motion visible.

The different particle masses matter. The same force acts on both nuclei, but the lighter nucleus experiences the greater acceleration because $a = F/m$. This is why the two velocity curves do not change at the same rate. Their momentum changes are equal in magnitude and opposite in direction, while their velocity changes depend on their masses.

Reading the velocity–time graph

The velocity–time graph shows how each nucleus responds throughout the encounter. A curve crossing zero means that nucleus is momentarily at rest in the lab frame. The point where the two curves meet is especially important: $v_1 = v_2$, so their relative velocity $v_2-v_1$ is zero. This corresponds to the minimum separation shown in the spatial view.

The graph remains in the lab frame even when a different reference frame is selected for the spatial simulation. This makes it possible to compare the original laboratory measurements with the relative motion seen by an observer moving with either particle.

Learning from different reference frames

In the Particle 1 frame, Particle 1 is held fixed on the display and only Particle 2's relative velocity is shown. In the Particle 2 frame, the roles are reversed. The displayed relative velocity is the difference between the two lab-frame velocities. Changing frames alters the description of the motion, but it does not alter measurable relationships such as the separation between the particles or the instant of closest approach.

This comparison is intended to build an important habit in mechanics: always identify the frame in which a velocity is measured. A particle may be moving in the lab frame while appearing stationary in a frame that moves with it. Relative velocity provides the link between these descriptions.

What to investigate

Try changing the masses, initial velocities, Coulomb strength, and initial separation. Increasing the repulsion strength generally produces a larger minimum separation. Increasing the approach speed gives the particles more kinetic energy, allowing them to move closer before reversing their relative motion. Changing either mass alters how the conserved total momentum is distributed between the two nuclei.

Use the timeline controls to compare three moments: when one particle first stops in the lab frame, when the particles reach their closest approach, and when the other particle stops. Then switch reference frames and observe how the same encounter can look different without changing the underlying physics.

The central lesson

The simulation connects four ideas that are often studied separately: Coulomb force, energy conversion, momentum conservation, and relative motion. Together they explain why the particles slow down, why they turn apart, how mass affects their response, and why velocity depends on the observer's frame of reference.