Calculate: the escape velocity for an atmospheric particle \(1000km\) above the surface of the earth, given that radius of earth is \(\left(\times10^{6}\right)m\) and acceleration due to gravity on surface of the earth \(=9.8\frac{m}{s^{2}}\).
Ans: \(v_{e}=10.42\frac{km}{s}\).
If, earth has mass nine times and radius be twice that of planet mars, Calculate: the velocity required by a rocket to pull out of the gravitational force of Mars. Take: Escape velocity at the surface of the earth to be \(11.2\frac{km}{s}\)
Ans: \(v=5.3\frac{km}{s}\).
Show that; the moon would depart forever, if its speed is increased by \(42\%\).
A black hole is a body from whose surface nothing may even escape. What is the condition for a uniform spherical body of mass \(M\) to be a black hole? What should be the radius of such a black hole if its mass is nine times of the mass of the earth. Given: \(M_{e}=\left(6\times10^{24}\right)kg\) and \(G=\left(6.67\times10^{-11}\right)Nm^{2}kg^{-1}\).
Ans: \(R=8cm\).
The radius of a planet is doubled that of the radius of the earth but their average densities are same. If, the escape velocities at the surface of the planet and at the surface of the earth are \(v_{p}\) and \(v_{e}\) respectively, then prove that \(v_{p}=2v_{e}\).
A body of mass \(200kg\) falls on the earth from infinity. What will be its velocity on reaching the earth? What will be its kinetic energy? Given: Radius of Earth\(=\left(6400km\right)\) and \(g=9.8\frac{m}{s^{2}}\). Air friction is negligible.
Ans: \(v=11.2\frac{km}{s}\) and \(K.E=\left(1.254\times10^{10}\right)J\).
A body is at a height equal to the radius of the earth from the surface of the earth. With what velocity the body may be thrown such that it goes out of the gravitational field of the earth? Given: \(M_{e}=\left(6.0\times10^{24}\right)kg\), \(R_{e}=\left(6.4\times10^{6}\right)m\) and \(G=\left(6.67\times10^{-11}\right)Nm^{2}kg^{-2}\).
Ans: \(v_{e}’=7.9\frac{km}{s}\).
A rocket starts vertically upwards with a speed \(v_{o}\). Show that its speed \(v\) at height \(h\) is given by \(v_{o}^{2}-v^{2}=\sqrt{\frac{2gh}{1+\frac{h}{R}}}\), where \(R_{e}\) is the radius of the earth.
The escape velocity of a projectile on the surface of the earth is \(11.2\frac{km}{s}\). A body is thrown upwards with thrice this velocity. If, the presence of the sun and other planets is ignored, then what is the velocity of the body far away from earth.
Ans: \(v_{f}=31.7\frac{km}{s}\).
An earth’s satellite makes a circle around the earth in \(90min\). How much is the height of the satellite above the surface of the earth? Given: Radius of the earth is \(6400km\) and \(g=9.8\frac{m}{s^{2}}\).
Ans: \(h=268km\).
A satellite with kinetic energy \(E_{k}[latex] is revolving around the earth in a circular orbit close to earth. How much more kinetic energy should be given to it so that it may just escape into outer space?
Ans: [latex]E_{k}\).
A body is projected vertically upwards with a velocity equal to the half of the escape velocity from the surface of the earth. What is the maximum height attained by the body?
Ans: \(h=\frac{R}{3}\).
In a two stage launch of a satellite to a height of \(150km\) and the second stage gives the necessary critical speed to put it into a circular orbit around the earth. Which stage requires more expenditure of fuel (neglecting, air resistance in the first stage.). Given: \(M_{e}=\left(6\times10^{24}\right)kg\) and \(R=6400km\) and \(G=\left(6.67\times10^{-11}\right)Nm^{2}kg^{-2}\).
Ans: \(W_{stage-2}>W_{stage-1}\).
The period of revolution of an artificial satellite just above the earth’s surface be \(T\) and density of earth \(\rho\). Prove That: \(\rho T^{2}\) is a universal constant. Given: \(G=\left(6.67\times10^{-11}\right)Nm^{2}kg^{-2}\).
A satellite orbits the earth at a height of \(500km\) from its surface. Calculate: its (i) Kinetic Energy (ii) Potential Energy (iii) Total Energy. Given: Mass of the satellite \(300kg\), Mass of the earth \(\left(6\times10^{24}\right)kg\), Radius of the earth \(\left(6.4\times10^{6}\right)m\) and \(G=\left(6.67\times10^{-11}\right)Nm^{2}kg^{-2}\).
Ans:(i) \(K.E=\left(8.7\times10^{9}\right)J\) (ii) \(U=-\left(17.4\times10^{9}\right)J\) and (iii) \(E=-\left(8.7\times10^{9}\right)J\).
Show that the velocity of a body released at a distance \(r\) from the centre of the earth, when strikes the surface of earth is given by \(v=\sqrt{2GM\left(\frac{1}{R}-\frac{1}{r}\right)}\), where \(R\) and \(M\) are the radius and mass of earth respectively. Also show that the velocity with which the meteorites strikes the surface of earth is equal to the escape velocity from the surface of earth.
Calculate: the energy required to move ab earth satellite of mass \(1000kg\) from a circular orbit of radius \(2R\) to that of radius \(3R\). Given: Mass of earth \(M_{e}=\left(5.98\times10^{24}\right)\), radius of the earth \(R_{e}=\left(6.37\times10^{6}\right)m\).
Two particles of equal mass \(m\) go around a circle of radius \(R\) under the action of their mutual gravitational attraction. What is the speed of each particle?
