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I have this exercise which I don't understand its solution.

I am attaching both the exercise and its solution.

It's problem 2: rocket trip.

What I don't understand in the solution they write that:

"and from this you get:

$$\omega < \frac{1}{R}\sqrt{1-2m/R}$$

Then plug in ##\omega = m/R^3## to obtain the possible radii for free falling motion,

$$R>3m$$

Now, if I plug in this ##\omega## I get: ##m/R^2<\sqrt{1-2m/R}## and squaring both sides and multiplying by R^4 I get: ##m^2<R^4-2mR^3## ; if ##R>3m## then ##R^4-2mR^3>27m^4## and this is greater than ##m^2## whenever ##m^2>1/27##; from what does this follow I don't see, can't ##m^2## be less than ##1/27##?

I am attaching both the exercise and its solution.

It's problem 2: rocket trip.

What I don't understand in the solution they write that:

"and from this you get:

$$\omega < \frac{1}{R}\sqrt{1-2m/R}$$

Then plug in ##\omega = m/R^3## to obtain the possible radii for free falling motion,

$$R>3m$$

Now, if I plug in this ##\omega## I get: ##m/R^2<\sqrt{1-2m/R}## and squaring both sides and multiplying by R^4 I get: ##m^2<R^4-2mR^3## ; if ##R>3m## then ##R^4-2mR^3>27m^4## and this is greater than ##m^2## whenever ##m^2>1/27##; from what does this follow I don't see, can't ##m^2## be less than ##1/27##?