EXERCISES

The escape speed of a projectile on the earth’s surface is $11.2 \mathrm{km} {\mathrm{s}}^{-1}$. A body is projected out with thrice this speed. What is the speed of the body far away from the earth? Ignore the presence of the sun and other planets. (Find the value in kmps upto one decimal place)

A satellite orbits the earth at a height of $400 \mathrm{km}$ above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? Mass of the satellite $=200 \mathrm{kg};$ mass of the earth  $=6.0\times {10}^{24} \mathrm{kg};$  radius  of the earth $=6.4\times {10}^6 \mathrm{m};$ $G=6.67\times {10}^{-11} {\mathrm{Nm}}^2{\mathrm{kg}}^{-2}$. 

Two stars each of one solar mass $\left(2\times {10}^{30} \mathrm{kg}\right)$ are approaching each other for a head on collision. When they are a distance ${10}^9 \mathrm{km}$, their speeds are negligible. What is the speed with which they collide? The radius of each star is ${10}^4 \mathrm{km}$. Assume the stars to remain undistorted until they collide. (Use the known value of $G$). 

Two heavy spheres each of mass $100 \mathrm{kg}$ and radius $0.10 \mathrm{m}$ are placed $1.0 \mathrm{m}$ apart on a horizontal table. What is the gravitational force and potential at the mid point of the line joining the centres of the spheres? Is an object placed at that point in equilibrium? If so, is the equilibrium stable or unstable?

As you have learnt in the text, a geostationary satellite orbits the earth at a height of nearly $36,000 \mathrm{km}$ from the surface of the earth. What is the potential due to earth’s gravity at the site of this satellite? (Take the potential energy at infinity to be zero). Mass of the earth$=6.0\times {10}^{24} \mathrm{kg},$ radius$=6400 \mathrm{km}$. 

A star $2.5$ times the mass of the sun and collapsed to a size of $12 \mathrm{km}$ rotates with a speed of $1.5$ revolution per second. (Extremely compact stars of this kind are known as neutron stars. Certain stellar objects called pulsars belong to this category). Will an object placed on its equator remain stuck to its surface due to gravity? (mass of the sun$=2\times {10}^{30} \mathrm{kg}$).

A spaceship is stationed on Mars. How much energy must be expended on the spaceship to launch it out of the solar system? Mass of the space ship$=1000 \mathrm{kg};$ mass of the sun$=2\times {10}^{30} \mathrm{kg};$ mass of mars$=6.4\times {10}^{23} \mathrm{kg};$ radius of mars$=3395 \mathrm{km};$ radius of the orbit of mars$=2.28\times {10}^8 \mathrm{km};$ $G=6.67\times {10}^{-11} {\mathrm{Nm}}^2{\mathrm{kg}}^{-2}$.

A rocket is fired ‘vertically’ from the surface of mars with a speed of $2 {\mathrm{kms}}^{-1}$. If $20\%$ of its initial energy is lost due to Martian atmospheric resistance, how far will the rocket go from the surface of Mars before returning to it? Mass of mars $=6.4\times {10}^{23} \mathrm{kg};$ radius of mars $=3395 \mathrm{km};$ $G=6.67\times {10}^{-11} {\mathrm{Nm}}^2{\mathrm{kg}}^{-2}$.