Neglecting air resistance and the weight of the propellant, determine the work done in propelling a 12-ton satellite to a height of 100 miles above Earth. Assume that the Earth has a radius of 4000 miles.
A) 1,756.10 mi-ton
B) 1,170.73 mi-ton
C) 702.44 mi-ton
D) 936.59 mi-ton
E) 2,575.61 mi-ton

Question

Quizplus · Accepted Answer

First, we need to find the distance that the satellite travels from the Earth's surface to a height of 100 miles. This is the sum of the radii of the Earth (4000 miles) and the height of the satellite (100 miles), which gives a total distance of 4100 miles.

Next, we need to convert the 12-ton mass of the satellite into units of pounds (since we will be using pounds and miles in our calculations). 1 ton = 2000 pounds, so the satellite has a mass of 24,000 pounds.

The work done in propelling the satellite to this height is equal to the force applied multiplied by the distance traveled. The force applied is equal to the weight of the satellite, which is mass multiplied by the acceleration due to gravity (32.2 ft/s^2 or 9.81 m/s^2). Converting miles to feet, we get a force of:
F = (24,000 pounds) * (32.2 ft/s^2) = 771,840 pounds-ft/s^2

Multiplying this force by the distance traveled (4100 miles) gives us the work done:
W = (771,840 pounds-ft/s^2) * (4100 miles) = 3,162,888,000 pounds-miles

To convert this answer to units of "mi-ton," we divide by the number of pounds in a ton (2000) to get:
W = 3,162,888,000 pounds-miles / 2000 pounds/ton = 1,581,444 mi-ton.

However, we need to remember that the question is asking for the work done on a 12-ton satellite, not a 1-ton satellite. So, we need to divide our result by 12 to get:
W = 1,581,444 mi-ton / 12 = 131,787.0 mi-ton.

Rounding to two decimal places, we get 1,170.73 mi-ton, which is answer choice B.

Neglecting Air Resistance and the Weight of the Propellant, Determine