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 Trajectory Calculation for Quadratic Drag of form -cv2

二次阻力形式为 -cv² 的竖直轨迹计算

Air drag will depend upon the area of the object in motion and the resulting trajectory will depend upon its mass. For that reason the sphere calculation is typically the starting point for modeling motion with quadratic drag. Departures from spherical geometry can then be tried by varying the drag coefficient.

空气阻力取决于物体的面积，而其轨迹则取决于其质量。因此，球体计算通常是处理二次阻力运动模型的起点。然后可以通过改变阻力系数来尝试偏离球面几何的情况。

But a trajectory can be modeled with just the general form -cv2 for the quadratic drag. This is an example of a vertical trajectory calculation.

但轨迹可以用一般的形式 -cv² 来建模，这适用于二次阻力的情况。这是一个垂直轨迹计算的例子。

Assume the object has a drag coefficient c = Ns2/m2 and a mass of m = gm.

假设物体的阻力系数为 c = Ns²/m²，质量为 m = 克。
中文译文中的待填/计算数值依次对应：1：c 2：m。实际数值以上方原输入框为准。

Under these conditions, it's terminal velocity will be
vt = m/s
vt = km/hr
vt = mi/hr

在这些条件下，它的终端速度将是 v_t = m/s，v_t = km/hr，v_t = mi/hr
中文译文中的待填/计算数值依次对应：1：vt 2：vtk 3：vtc。实际数值以上方原输入框为准。

and its characteristic time is
τ = s.

If this sphere is launched vertically with a velocity of
v0 = m/s = ft/s

如果这个球以初速度 v₀ = m/s = ft/s 垂直发射
中文译文中的待填/计算数值依次对应：1：v0 2：v0c。实际数值以上方原输入框为准。

then on the way up at height y1 = m = ft
it will have velocity v1 = m/s = ft/s

然后在上升过程中，当高度为 y₁ = m = ft 时，它将具有速度 v₁ = m/s = ft/s
中文译文中的待填/计算数值依次对应：1：y1 2：y1c 3：v1 4：v1c。实际数值以上方原输入框为准。

It will reach a peak height ypeak = m = ft
at time tpeak = s

它将在时间 t peak = s 达到峰值高度 y peak = m = ft
中文译文中的待填/计算数值依次对应：1：yp 2：ypc 3：tp。实际数值以上方原输入框为准。

On the way down at height y2 = m = ft
it will have velocity v2 = m/s = ft/s

在高度 y₂ = m = ft 下落时，其速度将为 v₂ = m/s = ft/s
中文译文中的待填/计算数值依次对应：1：y2 2：y2c 3：v2 4：v2c。实际数值以上方原输入框为准。

It will reach the ground at velocity
vimpact = m/s = ft/s = km/hr = mi/hr
at time timpact = s.

它将以速度 v 冲击地面，速度为 m/s = ft/s = km/hr = mi/hr，时间为 t impact = s。
中文译文中的待填/计算数值依次对应：1：vf 2：vfc 3：vfk 4：vfm 5：tf。实际数值以上方原输入框为准。

For comparison, if there were no air friction the projectile would have reached height
h = m = ft
and would impact with the original launch velocity at time t = s.

相比之下，若无空气摩擦，弹体将达到高度 h = m = ft，并在时间 t = s 时以原发射速度撞击。
中文译文中的待填/计算数值依次对应：1：h 2：hc 3：t2。实际数值以上方原输入框为准。
其特征时间是 τ = s。
中文译文中的待填/计算数值依次对应：1：tau。实际数值以上方原输入框为准。

Notes: This is not as practical a calculation form as the calculation for a sphere in air, but could be useful for obtaining an effective drag coefficient. If we were to model the trajectory of a 100 gram spherical rock of radius 2cm (density about 3000 kg/m3 or specific gravity 3) through air with density 1.29 kg/m3, that alternate form of the calculation would lead to a drag coefficient c = .0004 Ns2/m2. If you leave the values of c and radius at zero, they will default to the values for this rock. That will give you an example which you could then compare to the sphere calculation.

注：这种计算形式不如在空气中计算球体那样实用，但可能有助于获得一个有效的阻力系数。如果我们想模拟一个100克、半径2厘米（密度约3000 kg/m³或比重3）的球形岩石通过空气的轨迹，这种另一种计算形式将导致阻力系数c = 0.0004 Ns²/m²。如果你将c和半径的值留为零，它们将默认为该岩石的值。这将给你一个例子，你可以将其与球体计算进行比较。
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