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。

Freefall Velocity with Quadratic Drag

二次阻力下的自由下落速度

A freely falling object will be presumed to experience an air resistance force proportional to the square of its speed. The downward direction will be taken as positive, and the velocity as a function of time is the object of the calculation. The expressions will be developed for the two forms of air drag which will be used for trajectories:

一个自由下落的物体将被假定受到与速度平方成正比的空气阻力力。向下方向取为正，物体的 velocity 随时间的变化是计算的对象。将推导出两种空气阻力形式的表达式，这些表达式将用于计算轨迹：

although the first steps will be done with just the form -cv2 for simplicity.

尽管第一步将仅使用形式为 -cv 2 以简化处理。

The differential equation for the motion is

运动的微分方程是

which expresses the force in terms of the terminal velocity vt:

这表达了力与终端速度 v_t 之间的关系：

Integrating the motion equation yields

将运动方程积分得到

which expresses the fall time t in terms of the characteristic time for the motion

这表达了运动的特征时间与下落时间t之间的关系

The motion equation can then be solved for the velocity v:

运动方程可以解出速度 v：

If the falling object was released from rest at time t=0, the velocity expression becomes:

如果物体在时间t=0时从静止释放，速度表达式变为：

The nature of the motion is such that the speed is essentially at its terminal velocity vt after a few characteristic times. At t = 5τ, the speed is 0.99991 vt.

Freefall Distance as a Function of Time

时间作为自由下落距离的函数
运动的性质是，在几个特征时间后，速度基本上达到终端速度v_t。在t = 5τ时，速度为0.99991 v_t。

The distance of freefall y as a function of time may be obtained by integrating the velocity expression above.

自由下落的位移y作为时间的函数，可以通过上述速度表达式进行积分获得。

The form of this integral is

这个积分的形式是

The constant of integration C=0 because cosh(0)=1 and ln(1)=0, so the result of the integration is

Time of impact upon falling from peak height

从峰值高度下落时的撞击时间
常数积分常数C=0，因为cosh(0)=1且ln(1)=0，因此积分的结果为

If we substitute the peak height ypeak into the distance equation above so that it has reached the surface from which it was launched, we obtain the relationship

如果我们把峰值高度 y peak 代入上面的距离方程，使其达到发射点表面，我们得到以下关系：

Taking the exponential of both sides gives

and solving for the time yields the time of impact from ypeak:
对两边取指数得到，并解出时间得到碰撞时间从y峰值：

This expression is used in the vertical trajectory calculation.

这个表达式用于垂直轨迹计算。
 Index

Fluid friction

Reference
Fowles & Cassiday
Sec. 2.4
索引 流体摩擦 参考 Fowles & Cassiday 第2.4节
 
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