Preface

There are some details out of Examination-oriented education

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Physics

The second cosmic velocity

In class, we had learnt the first cosmic velocity, Now let’s quickly review.

Firstly, we do not consider the effects of rotation. (not consider the effects of the centrifugal force)

so there is mg=mv12Rmg=\frac{mv_1^2}{R} or mv12R=Gmm′R2\frac{mv_1^2}{R}=\frac{Gmm'}{R^2}

and we can get v1=gR=Gm′Rv_1=\sqrt{gR}=\sqrt{\frac{Gm'}{R}}

now we begin to analyze how can we get the second cosmic velocity

firstly, we know there should be gravity however the distance long.

but we can know that there is a place where the gravity will limits to infinity small and we can suppose there a object comes to the plant, so we can make a formula

flowing part I will use EgE_g express the gravitational potential energy

12mv2−Eg=0\frac{1}{2}mv^2-E_g=0

but what is the EGE_G?
it’s very easy, we can image a image, we plus the all EGE_G of the image to produce the total EGE_G, in this picture, the yellow area is the EGE_G we what to plus.

EG=∫R∞Gmm′R2dR=−Gmm′R∣R∞=Gmm′RE_G=\int_R^\infty \frac{Gmm'}{R^2}dR=-\frac{Gmm'}{R}\bigg|_R^\infty=\frac{Gmm'}{R}

and then there is the result.

v2=2Gm′R=v1v_2=\sqrt{\frac{2Gm'}{R}}=\sqrt{v_1}

now, you can also try to get the v3,v4,⋯ ,vnv_3, v_4,\cdots, v_n