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。

Force and Momentum

力与动量

The net external force acting on an object can be evaluated as the rate of change of momentum. This turns out to be a more fundamental way of stating the force than the use of Newton's second law. Using Newton's second law and momentum:

But this limited relationship can be generalized to
and further generalized by calculus methods to include instantaneous rates of change
This formulation of the force relationship permits varying mass, as in rocket propulsion.
这种力的关系式允许质量变化，如火箭推进。

In the language of calculus, we say that the force is defined as the derivative of the momentum. The process above involves the product rule for derivatives.

在微积分的语言中，我们说力是动量的导数。上述过程涉及导数的乘积法则。
作用在物体上的合外力可以表示为动量的变化率。这实际上比使用牛顿第二定律来表述力更为根本。利用牛顿第二定律和动量：但这种有限的关系可以通过微积分方法进一步推广，以包括瞬时变化率。
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Limits on Newton's 2nd law
牛顿第二定律的索引限制
 
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Rocket Propulsion

火箭推进
The thrust of a rocket can be modeled from a generalization of Newton's 2nd Law to include a variable mass:
In free space this would lead to a rocket velocity
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and for a rocket launched from the Earth's surface it leads to the expression
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at low altitudes where g can be considered to be constant.

(This is the launch of the Zvezda service module of the International Space Station on a Russian Proton-K rocket, July 12, 2000. Image source NASA.)
在太空中，这将导致火箭速度的产生。对于从地球表面发射的火箭，在低海拔地区，其中g可以视为常数，将得出表达式。（这张图显示了2000年7月12日，俄罗斯Proton-K火箭发射国际空间站的Zvezda服务舱。图片来源：NASA。）
火箭的推力可以从牛顿第二定律的一个推广形式来建模，该形式包括可变质量：
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Thrust of a Rocket

火箭推力
Rocket thrust results from the high speed ejection of material and does not require any medium to "push against". Conservation of momentum dictates that if material is ejected backward, the forward momentum of the remaining rocket must increase since an isolated system cannot change its net momentum.
NASA Space Shuttle launch
NASA 太空梭发射
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火箭推力来自于高速喷射物质的结果，并不需要任何介质来‘推动’。动量守恒定律指出，如果物质向后喷射，剩余火箭的前进动量必须增加，因为一个孤立系统不能改变其总动量。
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Rocket propulsion
索引 火箭推进
 
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Thrust of a Rocket

火箭推力
Rocket thrust
火箭推力
Show differential approach
展示微分方法
Further discussion of rocket thrust
火箭推力的进一步讨论
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Rocket propulsion
索引 火箭推进
 
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Thrust of a Rocket

火箭推力
Rocket thrust
火箭推力
Show algebra approach
展示代数方法
Further discussion of rocket thrust
火箭推力的进一步讨论
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Rocket propulsion
索引 火箭推进
 
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Rocket Thrust Discussion

火箭推力讨论
Developing the expression for rocket thrust involves the application of conservation of momentum in an accelerating reference frame. It can be developed in an approximate fashion using finite differences and algebra. This requires neglecting terms like ΔMΔv. When the appropriate limits are taken to get an instantaneous expression, these approximations disappear since the quantities dM and dt approach zero.
Show algebra approach
展示代数方法
Show calculus approach
展示微积分方法
推导火箭推力的表达式需要在加速参考系中应用动量守恒。可以近似地通过有限差分和代数方法推导出来。这需要忽略诸如ΔMΔv这样的项。当取适当的极限以得到瞬时表达式时，这些近似将消失，因为量dM和dt趋于零。
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Rocket propulsion
索引 火箭推进
 
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