Editing Envelope (mathematics) (section)

===Example 5===
[[File:Envelope cast.svg|thumb|The orbits' envelope of the projectiles (with constant initial speed) is a concave parabola. The initial speed is 10 m/s. We take ''g'' = 10 m/s<sup>2</sup>.]]
We consider the following example of envelope in motion. Suppose at initial height 0, one casts a [[Trajectory of a projectile|projectile]] into the air with constant initial velocity ''v'' but different elevation angles θ. Let ''x'' be the horizontal axis in the motion surface, and let ''y'' denote the vertical axis. Then the motion gives the following differential [[dynamical system]]:
:<math>\frac{d^2 y}{dt^2} = -g,\; \frac{d^2 x}{dt^2} = 0, </math>
which satisfies four [[initial condition]]s:
:<math>\frac{dx}{dt}\bigg|_{t=0} = v \cos \theta,\; \frac{dy}{dt}\bigg|_{t=0} = v \sin \theta,\; x\bigg|_{t=0} = y\bigg|_{t=0} = 0.</math>
Here ''t'' denotes motion time, θ is elevation angle, ''g'' denotes [[gravitational acceleration]], and ''v'' is the constant initial speed (not [[velocity]]). The solution of the above system can take an [[implicit function|implicit form]]:
:<math>F(x,y,\theta) = x\tan \theta - \frac{gx^2}{2v^2 \cos^2 \theta} - y = 0.</math>
To find its envelope equation, one may compute the desired derivative:
:<math>\frac{\partial F}{\partial \theta} = \frac{x}{\cos^2 \theta} - \frac{gx^2 \tan \theta}{v^2 \cos^2 \theta} = 0.</math>
By eliminating θ, one may reach the following envelope equation:
:<math>y = \frac{v^2}{2g} - \frac{g}{2v^2}x^2.</math>
Clearly the resulted envelope is also a [[concave function|concave]] [[parabola]].