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The force due to gravity on an object with mass m at a height h above the surface of the earth is

where R is the radius of the earth and t is the acceleration due to gravity for an object on the surface of the earth.

(a) Express F as a series in powers of h/R.

(b) Observe that if we approximate F by the first term in the series, we get the expression F ~ mt that is usually used when h is much smaller than R. Use the Alternating Series Estimation Theorem to estimate the range of values of h for which the approximation F ~ mg is accurate to within one percent. (Use R = 6400 km.)

mgR? F = (R + h)?