A body is projected from the Earth's surface with a velocity equal to half the escape velocity from that point at an angle ${45}^{\mathrm{o}}$ with horizontal. The maximum height attained by the body above the Earth's surface is (nearly)

Gravitational acceleration on the surface of a planet is $\frac{\sqrt{6}}{11}g$ , where g is the gravitational acceleration on the surface of the Earth. The average mass density of the planet is $\frac{2}{3}$ times that of the Earth. If the escape speed on the surface of the earth is taken to be $11 \mathrm{km} {\mathrm{s}}^{-1}$, the escape speed on the surface of the planet in $\mathrm{km} {\mathrm{s}}^{-1}$ will be

The particles each of mass $M$, are located at the vertices of an equilateral triangle of side $a$. At what speed must they move if they all revolve under the influence of their gravitational force of attraction in a circular orbit circumscribing the triangle while still preserving the equilateral triangle?