A rock of mass m* is shot radially upward fromthe surface of the earth. (* means

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A rock of mass m* is shot radially upward fromthe surface of the earth. (* means it is a dimensionalquantity.) R* is the radius of the earth and x*(t*) is theradial distance from the earths surface at time t* You can neglect air resistance. you are giventhe following equations and initial conditions d2x*/dt*2=(-g*R*2)/(x*+R*)2 x*(0)=0 dx*/dt*(0)=V* Now I am supposed to non dimenstionalize theseequations to get (where q is a dimensionless quantity) q(d2x/dt2)=-1/(1+x)2 x(0)=0 dx/dt=1 I understand that for example x*=x{x}, and t*=t{t}g*=g{x}/{t}2 R*=R{x) My understanding is you substitute the starredequations and the dimensional quantities cancel out giving me d2x/dt2=(-gR2)/(x+R)2for the first equation..which just seems like I proved that theequation is dimensionally correct. I dont understand how dx*/dt*(0)=V* changes todx/dt=1 either. I don't understand how to get to the other form(q(d2x/dt2)=-1/(1+x)2) . Ifeel like its simple and I am just missing something. I cannot seemto find and really simple examples online to help me,and we donthave a book for my class. If anyone could help me it would begreatly appreciated!!! A rock of mass m* is shot radially upward fromthe surface of the earth. (* means it is a dimensionalquantity.) R* is the radius of the earth and x*(t*) is theradial distance from the earths surface at time t* You can neglect air resistance. you are giventhe following equations and initial conditions d2x*/dt*2=(-g*R*2)/(x*+R*)2 x*(0)=0 dx*/dt*(0)=V* Now I am supposed to non dimenstionalize theseequations to get (where q is a dimensionless quantity) q(d2x/dt2)=-1/(1+x)2 x(0)=0 dx/dt=1 I understand that for example x*=x{x}, and t*=t{t}g*=g{x}/{t}2 R*=R{x) My understanding is you substitute the starredequations and the dimensional quantities cancel out giving me d2x/dt2=(-gR2)/(x+R)2for the first equation..which just seems like I proved that theequation is dimensionally correct. I dont understand how dx*/dt*(0)=V* changes todx/dt=1 either. I don't understand how to get to the other form(q(d2x/dt2)=-1/(1+x)2) . Ifeel like its simple and I am just missing something. I cannot seemto find and really simple examples online to help me,and we donthave a book for my class. If anyone could help me it would begreatly appreciated!!! x(0)=0 dx/dt=1 I understand that for example x*=x{x}, and t*=t{t}g*=g{x}/{t}2 R*=R{x) My understanding is you substitute the starredequations and the dimensional quantities cancel out giving me d2x/dt2=(-gR2)/(x+R)2for the first equation..which just seems like I proved that theequation is dimensionally correct. I dont understand how dx*/dt*(0)=V* changes todx/dt=1 either. I don't understand how to get to the other form(q(d2x/dt2)=-1/(1+x)2) . Ifeel like its simple and I am just missing something. I cannot seemto find and really simple examples online to help me,and we donthave a book for my class. If anyone could help me it would begreatly appreciated!!! d2x/dt2=(-gR2)/(x+R)2for the first equation..which just seems like I proved that theequation is dimensionally correct. I dont understand how dx*/dt*(0)=V* changes todx/dt=1 either. I don't understand how to get to the other form(q(d2x/dt2)=-1/(1+x)2) . Ifeel like its simple and I am just missing something. I cannot seemto find and really simple examples online to help me,and we donthave a book for my class. If anyone could help me it would begreatly appreciated!!!

Explanation / Answer

Let x =R           where is dimensionless distance. Multiplying it by theearth's radius gives it distance units

Sub into the original equation :d2x/dt2=(-g*R2)/(x+R)2

d2(R)/d(R/V)2 =(-g*R2)/(R + R)2

[R/(R/V)]*d2/d2 = -g*R2/[R2*(+1)2]

V*d2/d2 =-g/(+1)2

(V/g)*d2/d2 =-1/(+1)2

Let q = V/g

dx/dt(0)=V = d(R)/d(R/V) = V*(R/R)*d/d(0)= V*d/d (0)

V*d/d (0)= V

d/d (0)= V/V = 1

Also x(0) = 0 =>   R(0) = 0  => (0) = 0


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However are you sure this is the correct equation?

d2x*/dt*2=(-g*R*2)/(x*+R*)2

Does mass play any role? I am guessing that m*acceleration = force,and mass cancels.

Hope this helps

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