A rocket is fired with a speed upsilon=2sqrt(gR) near the earth.s surface and directed upwards. (a) Show that it will escape from the earth. (b) Show that in interstellar space its speed is upsilon=sqrt(2gR).

A rocket is fired with a speed upsilon=2sqrt(gR) near the earth.s surf

1.	A rocket is fired with a speed upsilon=2sqrt(gR) near the earth.s surface and directed upwards. (a) Show that it will escape from the earth. (b) Show that in interstellar space its speed is upsilon=sqrt(2gR).
Answer» Solution :(a) As PE of the rocket at the surface of earth is (-GMm/R) and at `oo`, zero, so energy required for escaping from earth `=0-((GMm)/R)=mgR[as g=(GM)/R^2]` And as initial KE of the rocket `1/2m upsilon^2=2mgR` is greater than the energy required for escaping (= mg R), the rocket will escape. (b) If `upsilon` isthe velocity of the rocket in interstellar SPACE (free from gravitational effects) then by CONSERVATION of energy, `1/2mn(2sqrt(gR))^2 -1/2m(sqrt(2gR))=1/2m upsilon^2` `upsilon^2=4gR to 2gR` or `upsilon=sqrt(2gR)`