Question

Neglecting air resistance and the weight of the propellant, determine the work (mi-tons) done in propelling a four-metric-ton satellite to a height of 220 miles above Earth. (Use 4,000 miles as the radius of Earth. Round your answer to two decimal places.)

Answer 1

the work done in propelling the satellite is 919.2 mile-tons

Explanation:

Given the data in the question;

we know that, the weight of a body varies inversely as the square of its distance from center of of earth;

⇒ F(x) = c / x²

given that F(x) = four-metric-ton = 4 × 1.102 = 4.408 tons

radius of earth = 4000 miles

we substitute

⇒ 4.408 = c / ( 4000 )²

c = 4.408 × ( 4000 )² = 70528000

so increment of work will be;

ΔW = ( 70528000 / x² ) Δx

Now work done in propelling to a height of 220 miles above Earth;

W = ₄₀₀₀∫⁴²²⁰ ( 70528000 / x² ) dx

W = 70528000[ - 1/x ]⁴²²⁰₄₀₀₀

W = 70528000[ - 1/4220 + 1/4000 ]

W = 70528000[ 1.3033175 × 10⁻⁵ ]

W = 919.2 mile-tons

Therefore, the work done in propelling the satellite is 919.2 mile-tons