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 interested space its speed is upsilon=sqrt(2gR)

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

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 interested space its speed is upsilon=sqrt(2gR)
Answer» Solution :(a) As PE of the ROCKET at the surface of earth of is `(-GMM//R)` and at `PROP` , zero , so energy required for escaping from earth. `=0-((GMm)/(R))=mgR["as " g=(GM)/R^2]` And is initial KE of the rocket `1/2mv^2 =2mgR`is greater than the energy required for escaping (=mgR) , the rocket will ESCAPE . (b) If `upsilon` is the 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))=1/2"m"upsilon^2` `upsilon^2=4gR-2gR" or " upsilon=sqrt(2gR)`