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。

Vertical Rocket

垂直火箭

Calculation

计算

Under the influence of the rocket thrust and gravity, the acceleration is

在火箭推力和重力的影响下，加速度是

Integrating from the initiation of the thrust to an arbitrary time t:

gives the velocity at time t:
从推力开始到任意时间t积分：得到时间t时的速度：
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Rocket propulsion
索引 火箭推进
 
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Rocket in Space

空间中的火箭
Under the influence of the rocket thrust in a gravity-free environment , the acceleration is

Calculation

计算

Integrating from the initiation of thrust to an arbitrary time t

从推力开始到任意时间t的积分

gives the velocity at time t:

给出时间t时的速度：

The above is the standard "rocket in space" scenario where you typically calculate the velocity after a given time of thrusting in terms of the amount of fuel burned and exhausted. If you want a specific expression for velocity in terms of time, that can be developed in terms of the exhaust rate as a function of time. If the burn rate is constant at Δm/Δt then the velocity as an explicit function of time would be

上述情况是标准的‘火箭在空间中’场景，通常会根据燃烧的燃料量和排出的燃料来计算推力作用一段时间后的速度。如果你想用时间来表达速度的特定表达式，可以基于时间函数的排气速率来推导。如果燃烧速率恒定为Δm/Δt，则速度作为时间的显式函数为

For a variable burn rate, the denominator would become an integral.

对于可变燃烧速率，分母将变为一个积分。
在无重力环境中，火箭推力引起的加速度是
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Rocket propulsion
索引 火箭推进
 
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Rocket in Space

空间中的火箭
Under the influence of the rocket thrust in a gravity-free environment, the resulting velocity can be calculated from:
First rocket equation


第一火箭方程
Second rocket equation


Develop expression

推导表达式
第二火箭方程
Velocity=Initial velocity + Exhaust velocity x ln(original mass /mass)
m0 = kg
m₀ = 千克
中文译文中的待填/计算数值依次对应：1：m0。实际数值以上方原输入框为准。
v0 = m/s
m = kg
v = m/s
The exhaust velocity
喷气速度
u = m/s
 To explore the effects of the various parameters on the motion of the rocket, you may substitute numbers in any of the boxes. When you have finished entering data, click on the active text above for the quantity you wish to calculate. The quantities will not be forced to be consistent until you click on the desired quantity. Note that the mass m0 includes the fuel mass, and the mass m can be considered to be the payload mass after a specified burn.
为探索各种参数对火箭运动的影响，您可以将数值输入任意一个框中。当您完成数据输入后，点击上方的活动文本以计算您希望计算的量。在点击您希望计算的量之前，这些量不会被强制保持一致。请注意，初始质量 m0 包括燃料质量，而质量 m 可以被认为是在指定燃烧后剩余的载荷质量。
Fowles and Cassiday give as example data for a satellite launch a low orbit velocity of about 8 km/s, an initial velocity of about 0.5 km/s from the rotation of the earth near the equator, and an exhaust velocity of 2.5 km/s. You could use this data above to confirm that you could get only about 5 percent of the total initial mass into orbit. They suggest that effective ejection speeds for most rocket fuels are on the order of 2 to 4 km/s. How much would a fuel on the upper end of that ejection speed range increase your payload?
在无重力环境中，火箭推力的影响下，所产生的速度可由以下公式计算：速度 = 初始速度 + 排气速度 × ln(原始质量 / 质量)。Fowles 和 Cassiday 给出一个卫星发射的例子数据：低轨道速度约为 8 km/s，地球赤道附近由于地球自转带来的初始速度约为 0.5 km/s，排气速度为 2.5 km/s。你可以用上述数据验证，仅能将总初始质量的约 5% 带入轨道。他们建议，大多数火箭燃料的有效喷射速度在 2 到 4 km/s 的范围内。如果燃料的喷射速度在该范围的上限，会增加多少有效载荷？
中文译文中的待填/计算数值依次对应：1：v0 2：v 3：u。实际数值以上方原输入框为准。
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Rocket propulsion
索引 火箭推进
 
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计算模块中未附着于文字的输入框：
中文译文中的待填/计算数值依次对应：1：m。实际数值以上方原输入框为准。