This sim plots the momentum that relativity assigns to a particle of rest mass m₀ against its speed v, next to Newton's m₀v. It shows where the two agree, where they part, and what controls the difference.

What's on screen

• Blue curve: the relativistic momentum p/c at each speed.
• Grey dashed line: Newton's momentum m₀v.
• Amber dashed line: the speed limit v = c.
• Violet marker: your point, set by the v slider.
• Controls panel: m₀, v, and the readout γ and p/c at your point. Hover the plot to read any other point.

Axes: v/c across (0 to 1), p/c up (0 to 10), both in units where c = 1.

Press ⌖ on a row to see where that control sits in the sim.

Fixed: the p/c axis stays at 0–10 in the full view and never rescales.

In Newton's mechanics momentum is just mass times speed:

So a steady push, which adds momentum at a steady rate (F = dp/dt), could take a particle past any speed. Relativity keeps F = dp/dt, so something else has to give way.

• The speed gives way. Along the blue curve the momentum keeps rising while v creeps toward the amber line without reaching it.
• The plot stops at p/c = 10. For m₀ = 1.0 the curve leaves the top at v = 0.995c, and the sim draws an arrow and "p → ∞".
• At the slider's top, v = 0.999c, p/c is already 22.3 for m₀ = 1.0: more than twice the height of the plot.

The blue curve is Newton's m₀v multiplied by the Lorentz factor γ, at every speed. Nothing switches on at some threshold.

• At v = 0.1c, γ = 1.005: a 0.5% change. That is the right edge of the low-velocity view, where blue and grey look like one line.
• The ratio layer shades p − m₀v, the momentum that γ adds. At v = 0.9c and m₀ = 1.0, p/c = 2.06 against Newton's 0.90, so the band, 1.16, is taller than Newton's whole value.

Momentum grows faster than speed, and the gap widens as v approaches c. Values for m₀ = 1.0:

• Doubling v from 0.45c to 0.90c multiplies p by 4.10. Newton gives 2.
• From 0.90c to 0.99c, v rises by 10% while p rises by a factor of 3.4. From 0.99c to 0.999c, v rises by 0.9% and p by a factor of 3.2.

Rest mass m₀ is a property of the particle, the same number in every frame. In p/c = γ m₀ β it multiplies everything, and γ contains no mass.

• Doubling m₀ doubles p at every speed. At v = 0.6c, p/c is 0.750 for m₀ = 1.0 and 1.50 for m₀ = 2.0.
• The speed limit stays at c, and γ does not change.
• The curve reaches the top of the fixed axis at a lower speed (0.995c for m₀ = 1.0, 0.981c for m₀ = 2.0), only because every value is larger.

Total momentum has to be conserved in a collision in every inertial frame. Velocities do not simply add between frames, so a collision that conserves Σ m₀v in one frame generally does not conserve it in a boosted frame. Σ γ m₀ v, together with the total energy, is conserved in every inertial frame. That is why γ belongs in the definition, and is not a fit to data. (The sim does not simulate collisions. This is the argument behind the curve.)

Energy and momentum together form one four-vector, with E = γ m₀c². With the metric η₀₀ = −1, η_ij = δ_ij, its length is the same in every frame:

In other frames p and E change, but m₀ does not. Moving the m₀ slider changes the particle, not the frame. Rearranged, the length is:

• Magnetic spectrometers. A charge q moving across a field B follows a circle of radius r = p/(qB), with p = γ m₀ v. Detectors measure momentum from the curvature of the track.
• The LHC. A 6.8 TeV proton has γ = E/(m₀c²) ≈ 7250, so its speed is within about 10⁻⁸ of c. The speed tells almost nothing, which is why beams are quoted in TeV and not in m/s.

Key points

• At low speed γ ≈ 1, and p matches Newton's m₀v.
• The rest mass multiplies the whole curve. γ carries no mass.
• γ is forced by conservation in every frame. With energy it makes a four-vector whose length, −m₀²c², is invariant.
• Near c, momentum (or energy) is the useful measure, not speed.

Common mix-ups

• "Momentum levels off near c." The speed levels off. p = γ m₀ v keeps growing; the 0–10 axis is where the plot stops, not where the physics does.
• "γ already carries the mass, so m₀ doesn't matter." γ = 1/√(1 − β²) contains no mass. Doubling m₀ doubles every p and leaves the speed limit at c.
• "Mass grows with speed, so m₀ is the wrong mass." The course uses one mass, the rest (invariant) mass. γ goes with the velocity.

Things to try on your own

• With m₀ = 1.0, find the speed at which p/c = 1.00, and compare it with Newton's.
• Set m₀ = 5.0 and find the speed at which the marker leaves the plot. Compare it with 0.995c for m₀ = 1.0.
• Turn the ratio layer on and find the speed above which the band is taller than the grey line at the same speed.