weirdfacts
Year 12 / Special relativity

Why a massive particle cannot reach c.

At speeds much lower than c, γ is close to one and p ≈ mv. At high speeds the correct momentum is γm₀v, where m₀ is rest mass. Increasing speed close to c requires increasingly large changes in momentum and energy.

A massive object cannot be accelerated to c with finite energy. Rest energy is m₀c² and changes in a system’s rest mass correspond to changes in its rest energy. The speed of a light signal remains c in every inertial frame, even when its source moves.

INTERACTIVE MODEL

Relativistic momentum

READ THE GRAPH

Explore the relationship

p = γm₀v; momentum grows without bound as a massive object approaches c.

Momentum / reference mc00.79861.5972.3963.19500.23750.4750.71250.95Speed / c
Momentum / reference mc

Tap or focus a plotted point to read its values.

View plotted data as a table
Explore the relationship
SeriesSpeed / cMomentum / reference mc
Momentum / reference mc00
Momentum / reference mc0.01583330.0158353
Momentum / reference mc0.03166670.0316826
Momentum / reference mc0.04750.0475537
Momentum / reference mc0.06333330.0634607
Momentum / reference mc0.07916670.0794159
Momentum / reference mc0.0950.0954316
Momentum / reference mc0.1108330.11152
Momentum / reference mc0.1266670.127695
Momentum / reference mc0.14250.143969
Momentum / reference mc0.1583330.160356
Momentum / reference mc0.1741670.17687
Momentum / reference mc0.190.193525
Momentum / reference mc0.2058330.210337
Momentum / reference mc0.2216670.227322
Momentum / reference mc0.23750.244496
Momentum / reference mc0.2533330.261876
Momentum / reference mc0.2691670.279481
Momentum / reference mc0.2850.297331
Momentum / reference mc0.3008330.315446
Momentum / reference mc0.3166670.333847
Momentum / reference mc0.33250.352559
Momentum / reference mc0.3483330.371607
Momentum / reference mc0.3641670.391016
Momentum / reference mc0.380.410817
Momentum / reference mc0.3958330.43104
Momentum / reference mc0.4116670.451719
Momentum / reference mc0.42750.47289
Momentum / reference mc0.4433330.494595
Momentum / reference mc0.4591670.516876
Momentum / reference mc0.4750.539781
Momentum / reference mc0.4908330.563365
Momentum / reference mc0.5066670.587684
Momentum / reference mc0.52250.612803
Momentum / reference mc0.5383330.638795
Momentum / reference mc0.5541670.665741
Momentum / reference mc0.570.693731
Momentum / reference mc0.5858330.722866
Momentum / reference mc0.6016670.753263
Momentum / reference mc0.61750.785053
Momentum / reference mc0.6333330.818388
Momentum / reference mc0.6491670.853441
Momentum / reference mc0.6650.890414
Momentum / reference mc0.6808330.929544
Momentum / reference mc0.6966670.971106
Momentum / reference mc0.71251.01543
Momentum / reference mc0.7283331.06291
Momentum / reference mc0.7441671.11403
Momentum / reference mc0.761.16937
Momentum / reference mc0.7758331.22965
Momentum / reference mc0.7916671.29579
Momentum / reference mc0.80751.36895
Momentum / reference mc0.8233331.45066
Momentum / reference mc0.8391671.54294
Momentum / reference mc0.8551.64858
Momentum / reference mc0.8708331.7715
Momentum / reference mc0.8866671.91749
Momentum / reference mc0.90252.09548
Momentum / reference mc0.9183332.32015
Momentum / reference mc0.9341672.61791
Momentum / reference mc0.953.04243
Momentum / reference mc: 0

Explore: Compare the low-speed part with the rapid increase close to c.

p=γm0vE0=m0c2p=\gamma m_0v\qquad E_0=m_0c^2
WORKED EXAMPLE

Compare relativistic momentum with m₀v at 0.60c.

  1. γ = 1.25.
  2. Relativistic momentum is 25% greater than m₀v.
Assumed knowledge

Lorentz factor and momentum.

Learning checkpoints & sourceYOUR LEARNING CHECKPOINT
  • Use relativistic momentum.
  • Connect rest mass and energy.
QCAA Physics · Unit 4 · Special relativity ↗
READY TO TRY IT?

Put the idea to work.

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