Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Relativistic Mechanical Quantities

相对论力学量

A number of ordinary mechanical quantities take on a different form as the speed approaches the speed of light.

当速度接近光速时，许多普通力学量的形式会有所不同。


At what energies must relativistic expressions be used?
在什么能量下必须使用相对论表达式？
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Relativistic Momentum

相对论动量

The relativistic momentum is given by

相对动量由下式给出

which is the ordinary definition of momentum with the mass replaced by the relativistic mass.

这是将质量替换为相对论质量的普通动量定义。
For v =
对于 v =
c, gamma = , and rest mass
c，伽马 = ，以及静止质量
中文译文中的待填/计算数值依次对应：1：gam。实际数值以上方原输入框为准。
m0 = x10^kg = me= mp
m₀ = x10⁴ kg = mₑ = mₚ
中文译文中的待填/计算数值依次对应：1：m0b 2：m0p 3：mme 4：mmp。实际数值以上方原输入框为准。
m0 = MeV/c2 = GeV/c2
m₀ = MeV/c² = GeV/c²
中文译文中的待填/计算数值依次对应：1：mmev 2：mgev。实际数值以上方原输入框为准。

the relativistic momentum is

相对论动量是
p =x10^kg m/s = MeV/c = GeV/c
10^ kg m/s = MeV/c = GeV/c
中文译文中的待填/计算数值依次对应：1：pb 2：pp 3：pmev 4：pgev。实际数值以上方原输入框为准。
compared to the non-relativistic result p = mv = x10^kg m/s
which would then be in error by %.

In the above calculations, one of the ways of expressing mass and momentum is in terms of electron volts. It is typical in high energy physics, where relativistic quantities are encountered, to make use of the Einstein relationship to relate mass and momentum to energy. In relativistic mechanics, the quantity pc is often used in momentum discussions. It has the units of energy.

在上述计算中，质量与动量的一种表达方式是用电子伏特来表示。在高能物理中，当遇到相对论量时，通常利用爱因斯坦关系将质量与动量与能量联系起来。在相对论力学中，动量讨论中经常使用量pc。它具有能量的单位。
For extreme relativistic velocities where
在极端相对论性速度的情况下，其中
then
然后

A useful application of the quantity pc is in the calculation of the velocity as a fraction of c.

动量的量pc在计算速度与光速之比时有重要作用。
and as
以及

with the limiting case applying for the momentum of a photon.
Momentum of photon
光子动量
Usefulness of the quantity pc
量pc的实用性
与非相对论结果p = mv = x10^ kg m/s相比，这将导致误差为%。在光子动量的极限情况下适用。
中文译文中的待填/计算数值依次对应：1：v2 2：pnb 3：pnp 4：per。实际数值以上方原输入框为准。
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Momentum of Photon

光子的动量

For a photon, the relativistic momentum expression

approaches zero over zero, so it can't be used directly to determine the momentum of a zero rest mass particle. But the general energy expression can be put in the form

and by setting rest mass equal to zero and applying the Planck relationship, we get the momentum expression:

对于光子来说，相对论动量表达式在零除的情况下趋于零，因此不能直接用来确定零静止质量粒子的动量。但一般能量表达式可以改写为形式，并通过将静止质量设为零并应用普朗克关系，我们得到动量表达式：
Wave-particle duality application
波粒二象性应用
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Relativistic Energy in Terms of Momentum

相对动量中的能量

The famous Einstein relationship for energy


can be blended with the relativistic momentum expression

to give an alternative expression for energy.
The combination pc shows up often in relativistic mechanics. It can be manipulated as follows:

and by adding and subtracting a term it can be put in the form:

which may be rearranged to give the expression for energy:

Note that the m with the zero subscript is the rest mass, and that m without a subscript is the effective relativistic mass.
爱因斯坦关于能量的著名关系式可以与相对论动量表达式结合，从而得到能量的另一种表达式。动量的乘积pc在相对论力学中经常出现。它可以这样操作：通过加减一个项，可以将其表示为：这可以重新排列为能量的表达式：请注意，带有零下标的m是静止质量，而没有下标的m是有效的相对论质量。
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