yet another doubt about gravitation. we must affirm that it's also not a homework question, so, we won't need any number, only the formulas.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

1)suppose that an object is located in a distance r from the center of the Earth (not necessary to say that r is larger than the Earth's radius). we want to know the formula of the speed necessary (launching it horizontally) for that object to describe(A)a circular orbit of radius r.

we additionally want to know how to make this object describe(B)an elliptical,(C)a parabolic and a(D)hyperbolic orbit.

2)we would also like to know how to calculate the speed necessary to launch an object from the surface of the Earth, so it can enter in a circular orbit of radius r (bigger than the radius of the Earth).

2. Relevant equations

gravitational force:

F_{g}= GmM/r², where G is the gravitational constant, m is the mass of the object, M is the mass of the Earth and r is the distance between the center of the Earth and the center of the object (in this case, the distance r mentioned above).

centripetal force:

F_{cp}= mv²/r (where v is the speed of the object).

conservation of energy

[tex]\frac{mv^{2}_{0}}{2}-\frac{GmM}{R} = \frac{mv^{2}_{1}}{2}-\frac{GmM}{r}[/tex], where R is the radius of the Earth and r is the distance we mentioned above.

3. The attempt at a solution

1) (A)for the object to describe a circular orbit, its centripetal force has to balance with the gravitational force:

F_{cp}= F_{g}

mv²/r = GmM/r²

v²/r = GM/r²

v² = GM/r

[tex]v = \sqrt{\frac{GM}{r}}[/tex]

for B, C and D, we have no clue, because we don't know the mathematical condition to obtain those orbits.

NOTE: does v have to be horizontal? why wouldn't an object thrown with vertical v = [tex]v = \sqrt{\frac{GM}{r}}[/tex] also respect that F_{cp}= F_{g}?

2)we don't have much idea where to start, but we think that this velocity will have a vertical component and a horizontal component which is [tex]v_x = \sqrt{\frac{GM}{r}}[/tex].

the vertical velocity would be v_{0y}: [tex]\frac{mv^{2}_{0y}}{2}-\frac{GmM}{R} = \frac{mv^{2}_{1y}}{2}-\frac{GmM}{r}[/tex].

in this case, we think that v_{1y}would have to be zero.

thank you advance.

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# Finding the minimum speed for putting an object in orbit

Can you offer guidance or do you also need help?

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