Question

how to show that a body projected from the earth with a speed of square root of 2gR will never return?????. Radius of the Earth is R.AND State two assumptions for this result.

Answer 1

Consider that you want to project the body from the surface of the Earth to infinity, say from h=0 to infinity. You need to know the required potential energy and with that calculate the kinetic energy (velocity) required.

The gravitational force on the surface is:\[F_E=G \frac{ M_E ·m}{ R_E^2 }=g_0·m\]and the gravitational force at a distance h is:\[F_h=G \frac{ M_E·m }{ (R_E+h)^2 }\]. Then we can say:\[F_h=F_E \frac{ R_E^2 }{ (R_E+h)^2 }=m·g_0·\frac{ R_E^2 }{ (R_E+h)^2 }\]The required Energy to move the body from h=0 to infinity is:\[E=m·g_0·R_E^2\int\limits_{0}^{\infty}\frac{ dh }{ (R_E+h)^2 }=m·g_0·R_E^2\left[ \frac{ 1 }{ R_E+h } \right]_\infty^{0}=m·g_0·R_E\]This energy has to equal the kinetic energy:
\[m·g_0·R_E=\frac{ 1 }{ 2 }m·v^2\rightarrow v=\sqrt{2·g_0·R_E}\]

Answer 2

In a nutshell, a velocity of sqrt(2gR) is enough to provide an energy equivalent to the energy required to move the body from the surface of Earth to infinity. It is called escape velocity. Similar reasoning is used to calculate the radius of black holes (Schwarzschild's radius) assuming escape velocity is that of light (c)

Answer 3

"Welcome to OpenStudy. I can answer your questions or guide you. Please use the chat for off topic questions. And remember to give the helper a medal, by clicking on "Best Answer". We follow a code of conduct, ( http://openstudy.com/code-of-conduct ). Please take a moment to read it."