When does relativistic momentum matter?
When v > ~0.1c (γ > ~1.005). At LHC energies (99.999% c), γ ≈ 7500 and relativistic momentum is 7500× the classical value.
Relativistic Momentum Calculator
p = γmv
Calculate relativistic momentum p = γmv, where γ = 1/√(1−v²/c²). Important at speeds near c.
Calculate
Enter values above to calculate
How it calculates
1
c = 299,792,458 m/s (exact)
2
γ = 1/√(1 − v²/c²) (Lorentz factor)
3
p = γ × m₀ × v
Formula
p = γmv
Variable Table
| Symbol | Quantity | SI Unit |
|---|---|---|
| p | Relativistic momentum | kg·m/s |
| γ | Lorentz factor | dimensionless |
| m₀ | Rest mass | kg |
| v | Velocity | m/s |
| c | Speed of light = 2.998×10⁸ | m/s |
How to Use This Calculator
This relativistic momentum calculator is built for quick physics checks and worked-problem review. Enter values in the units shown beside each input, then compare the result with the formula and variable table before using it in a longer solution. The calculator does the arithmetic, but the physics still depends on choosing a model that matches the situation.
Start by identifying the system, the known quantities, and the quantity you want to find. If a value is given in a non-SI unit, convert it before substitution. A correct numerical answer with mixed units can still be physically wrong, especially when squared units, inverse seconds, charges, temperatures, or distances are involved.
Assumptions and Limits
The formula p = γmv is a model, not a universal description of every possible case. It assumes the quantities in the variable table are the relevant quantities for the problem and that hidden effects are either negligible or already included in the inputs. If friction, drag, relativistic speeds, changing fields, non-constant temperature, or geometry-specific effects matter, check whether a more complete model is needed.
Use the result as a magnitude and units check. Ask whether the answer has the right sign, whether it grows or shrinks when an input changes, and whether the limiting cases make sense. Setting an input to zero, doubling a quantity, or using a very large value is often enough to catch a formula choice or unit mistake before it reaches a final answer.
Worked Example
Proton (m = 1.67×10⁻²⁷ kg) at v = 0.9c. Find relativistic momentum.
Step 1: γ = 1/√(1−0.81) = 1/√0.19 ≈ 2.294
Step 2: p = 2.294 × 1.67×10⁻²⁷ × 0.9 × 3×10⁸
Answer: p ≈ 1.04×10⁻¹⁸ kg·m/s (classical would give 4.5×10⁻¹⁹)
Common Mistakes
- At v = 0.1c, relativistic correction is only 0.5% — use classical below ~0.1c
- v cannot reach or exceed c for massive objects
- As v → c, γ → ∞, so p → ∞ — this is why c is an absolute speed limit
Related
Frequently Asked Questions
Why can nothing reach c?
As v → c, γ → ∞, so p → ∞. Accelerating a massive object to c would require infinite energy.