The gravitational force on a rocket depends on the altitude of the rocket and it

Question

The gravitational force on a rocket depends on the altitude of the rocket and its mass. The formula for the gravitational force (in newtons, for you metric system freaks) is F=398755000 m/(6378+r)^2 where m is the mass of the rocket in kilograms and r is the distance above the earth's surface in kilometers. The rocket weighs 400000 kg, and uses up fuel at the constant rate of 220 kg per second during the first five minutes. After two minutes, its altitude is 60 km and increasing at 1.8 km per second. At what rate is the gravitational force on the rocket decreasing at this time?

Explanation / Answer

F = 398755000 m / (6378 + r)2

taking derivative both side relative to "t"

dF/dt = (398755000 m) (-2 / (6378 + r)3 ) (dr/dt) + 398755000/ (6378 + r)2 ) (dm/dt)

m = 400,000 kg , dm/dt = - 220 kg/sec      ,   r = 60 km = 60,000 m , dr/dt = 1800 m/s

dF/dt = ((398755000 (400,000)) (-2 / (6378 + 60000)3 ) (1800) ) + 398755000/ (6378 + 60000)2 ) (- 220)

dF/dt = - 1983.25 N/s

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