The Work Done Against Gravity in Propelling an Object with Mass

Essay

The work done against gravity in propelling an object with mass m kg to an altitude of h m above the surface of the earth is given by $W = \int _ { 637 \times 10 ^ { 6 } } ^ { 637 \times 10 ^ { 6 } + h } \frac { G M _ { E } m } { r ^ { 2 } } d r$ where 6.37 $\times$ 10⁶ m is the radius of the earth, $G \approx 6.667 \times 10 ^ { - 11 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { kg } ^ { 2 }$ and $M _ { E } \approx 5.90 \times 10 ^ { 24 } \mathrm {~kg}$ is the mass of the earth.(a) Find the work required to launch a 1000-kilogram satellite vertically to an altitude of 1000 km.(b) The formula shows that the work is dependent on h. Show that $\int _ { 637 \times 10 ^ { 6 } } ^ { \infty } \frac { G M _ { Z } m } { r ^ { 2 } } d r$ is convergent. What is the physical significance of the fact that this improper integral is finite?