A particle is projected from point ${}^{\prime }{A}^{\prime }$ , that is at a distance $4R$ from the centre of the Earth, with speed $V_0$ as shown. If the particle passes grazing the earth's surface, then it is found that $V_0=4000\times \sqrt{N}$. Find the value of N.
[Take $\frac{GM}{R}=6.4\times {10}^7\frac{{\text{m}}^2}{{\text{s}}^2}$]

Q. A particle is projected from a point A that is at a distance of $4R$ from the centre of the earth with speed $V_1$ in a direction making ${30}^{\circ }$ with the line joining the centre of the earth and point A as shown in figure. Consider the gravitational interaction only between these two bodies (use $\frac{GM}{R}=6.4\times {10}^7{m}^2/s^2$). The speed $V_1$ if particle passes grazing the surface of the earth is:

Q. A particle is projected from a point A that is at a distance of $4R$ from the centre of the earth with speed $V_1$ in a direction making ${30}^{\circ }$ with the line joining the centre of the earth and point A as shown in figure. Consider the gravitational interaction only between these two bodies (use $\frac{GM}{R}=6.4\times {10}^7{m}^2/s^2$). The speed $V_1$ if particle passes grazing the surface of the earth is:

Q. A body is projected from point $\text{A}$, that is at a distance $4R$ from the centre of the earth with speed $v_1$, in a direction making angle ${30}^{\circ }$ with the extension of line joining centre of the earth and point $\text{A}$. Find the speed $v_1\left(\text{in m/s}\right)$ of the body if the body passes earth by grazing its surface.
$[$ Neglect air resistance and rotation of earth, and use $\frac{GM}{R}=6.4\times {10}^7{\text{m}}^2/{\text{s}}^2$ for calculation. $]$

Q. For a particle projected in a transverse direction from a height h above earth's surface, find the minimum initial velocity so that it just grazes the surface of earth such that path of this particle would be an ellipse with centre of earth as the farther focus, point of projection as the apogee and a diametrically opposite point on earth's surface as perigee.

Q. A particle of mass $m$ is projected from the surface of earth with a speed $V_0$ ($V_0$ < escape velocity). The speed of the particle at a height h$=$R (radius of the earth) is