If $M$ is the mass of a planet and $R$ is its radius,then in order to become a black hole (where $c$ is the speed of light),which condition must be satisfied?

Earth has mass $M_1$ and radius $R_1$,and the moon has mass $M_2$ and radius $R_2$. The distance between their centers is $r$. $A$ body of mass $M$ is placed on the line joining them at a distance $r/3$ from the center of the Earth. To project the mass $M$ to escape to infinity,the minimum speed required is:

$A$ small asteroid is orbiting around the sun in a circular orbit of radius $r_0$ with speed $v_0$. $A$ rocket is launched from the asteroid with speed $v = \alpha v_0$,where $v$ is the speed relative to the sun. The highest value of $\alpha$ for which the rocket will remain bound to the solar system is (ignoring gravity due to the asteroid and effects of other planets).

$A$ body is projected vertically from the surface of the earth of radius $R$ with a velocity equal to half of the escape velocity. The maximum height reached by the body is

For a planet having mass equal to the mass of the Earth but a radius that is one-fourth of the radius of the Earth,the escape velocity for this planet will be (in $km/s$): (in $km/s$)

The escape speed of an object on the surface of the earth is $V$. If the object is thrown out with speed $4V$ from the surface of the earth,what will be the speed of the object far away from the earth?