A satellite of mass is raised straight up from the Earth's surface

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Accumulation: integration and the techniques that make it possible

Problem

A satellite of mass

m = 500\,\text{kg}

is raised straight up from the Earth's surface, radius

R_E = 6.37\times10^6\,\text{m}

. Beyond the surface the gravitational acceleration falls with distance

r

from the Earth's centre as

g(r) = g\,\frac{R_E^2}{r^2}, \qquad r \ge R_E,

where

g = 9.81\,\text{m s}^{-2}

is the acceleration at the surface. Air resistance is ignored throughout. ((a)) Write the work needed to raise the satellite from the surface to a radius

b

as a definite integral, and evaluate it. ((b)) Take

b \to \infty

to find the work needed to raise the satellite to an unbounded distance, stating this as the limit that defines the improper integral.

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