A rocket is fired with a speed v = 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 it speed is v = sqrt(2gR).

A rocket is fired with a speed v = 2sqrt(gR) near the earth's surface

1.	A rocket is fired with a speed v = 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 it speed is v = sqrt(2gR).
Answer» Solution :(a) As PE of the rocket at the surface of the 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)/(2)mv^(2) = 2mgR` isgreater than the energy required for escaping (=MG R), the rocket will escape. (b) If `v` is the VELOCITY of the rocket in interstellar space (free from gravitational effects) then by conservation of energy. `(1)/(2)mn(2sqrt(gR))^(2) - (1)/(2)m(sqrt(2gR)) = (1)/(2) m v^(2)` `v^(2) = 4gR - 2gR` or `v = sqrt(2gR)`