The second law of Newton and the universal gravitation law allows to find the results for the question about the dependencies of the speed in the movement of the satellite are:
a) The velocity does not depend on the mass of the satellite.
b) The velocity depends on the inverse of the square root of the distance.
c) The velocity is proportional to the square root of the Earth's mass.
The Law of Universal Gravitation indicates that the force is proportional to the masses of the body and inversely proportional to the square of its distance,
[tex]f = - G \frac{Mm}{r^2}[/tex]
Where F is the force, G the universal gravitational constant, M and m the mass of the two bodies, r the distance between the bodies, see attached.
Let's apply Newton's second law to motion.
F - ma
[tex]- G \frac{Mm}{r^2 } = m \ a[/tex]
a = [tex]- G \frac{M}{r^2 }[/tex]
The acceleration is centripetal, which is why it is given by the expression.
[tex]a = \frac{v^2}{r}[/tex]
let's substitute.
[tex]\frac{v^2}{r} = G \frac{M}{r^2}[/tex]
v² = [tex]G \frac{M}{r}[/tex]
We can see that the acceleration does not depend on the mass of the satellite and with centripetal accelerations, the speed does not depend on the mass of the satellite either.
The speed expression this depends inversely with the square root of the distance.
[tex]v= \sqrt{\frac{GM}{r} }[/tex]
In this expression the speed depends on the square root of the Earth's mass.
In conclusion, using Newton's second law and the law of universal gravitation we can find the results for the question about the dependencies of the speed in the movement of the satellite are:
a) The speed does not depend on the mass of the satellite.
b) The velocity depends on the inverse of the square root of the distance.
c) Velocity is proportional to the square root of the Earth's mass.
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