the square (

the square ( l/ is G m U r l . ESCAPE SPEED If a stone is thrown by hand, we see it falls back to the earth. Of course using machines we can shoot an object with much greater speeds and with greater and greater initial speed, the object scales higher and higher heights. A natural query that arises in our mind is the following: ‘can we throw an object with such high initial speeds that it does not fall back to the earth?’ The principle of conservation of energy helps us to answer this question. Suppose the object did reach infinity and that its speed there was V f . The energy of an object is the sum of potential and kinetic energy. As before W denotes that gravitational potential energy of the object at infinity. The total energy of the projectile at infinity then is f W ∞= ( . ) If the object was thrown initially with a speed V i from a point at a distance ( h + R E ) from the centre of the earth ( R E = radius of the earth), its energy initially was GmM E h W ( . ) Fig. . By the principle of energy conservation Eqs. ( . ) and ( . ) must be equal. Hence f GmM ( . ) The R.H.S. is a positive quantity with a minimum value zero hence so must be the L.H.S. Thus, an object can reach infinity as long as V i is such that GmM ≥ ( . ) The minimum value of V i corresponds to the case when the L.H.S. of Eq. ( . ) equals zero. Thus, the minimum speed required for an object to reach infinity (i.e. escape from the earth) corresponds to min GmM m V ( . ) If the object is thrown from the surface of the earth, h = , and we get