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。

Time to Peak for Quadratic Drag

二次阻力下的峰值时间

A vertically launched object will be presumed to experience an air resistance force proportional to the square of its speed. For a launch speed of v0 the object is to calculate the time to the peak. The expressions will be developed for the two forms of air drag which will be used for trajectories:

竖直发射的物体将被假定受到一个与速度平方成正比的空气阻力力。对于初速度为v₀的发射，需计算到达顶点的时间。将推导两种形式的空气阻力表达式，这些表达式将用于轨迹计算：

but the simpler -cv2 form will be used initially for simplicity and the forms for terminal velocity vt and characteristic time τ will be used.

但为了简化，最初将使用更简单的 -cv² 形式，终端速度 v_t 和特征时间 τ 的形式将被使用。

The motion equation for this vertical launch is

这种垂直发射的运动方程是

which can be integrated for time t in the form

可以以时间 t 的形式进行积分

but this is a non-trivial integral. Using an integral table it can be expressed as

但这是一个非平凡的积分。利用积分表可以将其表示为

We finally reach an expression for the velocity as a function of time:

我们终于得到了一个速度作为时间函数的表达式：

Now to find the time at the peak of the vertical trajectory, we can set the velocity equal to zero, yielding

现在要找到垂直轨迹的顶点时刻，我们可以将速度设为零，得到

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
A. Douglas Davis
Sec. 2.6
索引 流体摩擦 参考 A. Douglas Davis 第2.6节
 
HyperPhysics***** Mechanics ***** Fluids
HyperPhysics ***** 力学 ***** 流体
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Peak Height for Quadratic Drag

二次阻力的峰值高度

A vertically launched object will be presumed to experience an air resistance force proportional to the square of its speed. For a launch speed of v0 the object is to calculate the peak height. The expressions will be developed for the two forms of air drag which will be used for trajectories:

一个竖直发射的物体将被假设受到一个与速度平方成正比的空气阻力力。对于初速度为v₀的物体，需要计算其最高高度。将推导两种形式的空气阻力表达式，这些表达式将用于轨迹计算：

but the simpler -cv2 form will be used initially for simplicity and the forms for terminal velocity vt and characteristic time τ will be used.

但为了简化，最初将使用更简单的 -cv² 形式，终端速度 v_t 和特征时间 τ 的形式将被使用。

Since we have an expression for velocity as a function of time from above,

由于我们上面已经得到了速度作为时间函数的表达式，

it can be integrated to find an expression for distance y

它可以被积分以得到距离 y 的表达式

Again, a good integral table is needed to obtain an expression for y:

再次，需要一个良好的积分表来获得y的表达式：

Using the expression for velocity above, it is found that at the peak where v=0:

利用上述速度的表达式，可以发现，在速度为零的顶点处：

The expression for the peak height is then

然后是峰值高度的表达式

This expression for ypeak is used in the vertical trajectory calculation.

这个表达式用于垂直轨迹计算
 Index

Fluid friction

Reference
A. Douglas Davis
Sec. 2.6
索引 流体摩擦 参考 A. Douglas Davis 第2.6节
 
HyperPhysics***** Mechanics ***** Fluids
HyperPhysics ***** 力学 ***** 流体
R Nave
 Go Back
返回