A rocket lifts off the pad at Cape Canaveral. According to Newton's Law of Gravitation, the force of gravity on the rocket is given by
F(x) = − (GMm/x^2)
where M is the mass of the earth, m is the mass of the rocket, G is a universal constant, and x is the distance (in miles) between the rocket and the center of the earth. Take the radius of the earth to be 4000 miles, so that x > 4000 miles.
Find the work, W1, done against gravity when the rocket rises 1800 miles.
W1 =
GMm (mile-pounds)
Next, find the limit of the work, W2, as the rocket rises infinitely far from the earth.
W2 =
GMm (mile-pounds)
To find the work done against gravity when the rocket rises 1800 miles, we can use the formula for work:
W = ∫ F(x) * dx
Since the force of gravity is given by F(x) = -GMm/x^2, we substitute this into the work formula:
W1 = ∫ (-GMm/x^2) * dx
Integrating this expression will give us the work done against gravity. To do this, we can rewrite the expression as:
W1 = ∫ (-GMm * x^(-2)) * dx
To integrate this, we add 1 to the exponent and divide by the new exponent:
W1 = ∫ (-GMm * x^(-1)) / (-1) * dx
Simplifying further:
W1 = -GMm * (-x^(-1)) = GMm/x
Now, we can evaluate this expression over the given range of x, from 4000 miles to 5800 miles (1800 miles above the Earth's radius):
W1 = GMm/x |(from 4000 to 5800)
Substituting the values into the formula:
W1 = [GMm/5800] - [GMm/4000]
Simplifying:
W1 = GMm * (1/5800 - 1/4000)
= GMm * (4000 - 5800)/(4000 * 5800)
Therefore, the work done against gravity when the rocket rises 1800 miles is given by:
W1 = GMm * (1/5800 - 1/4000) (mile-pounds)
To find the limit of the work as the rocket rises infinitely far from the Earth (W2), we need to evaluate the work expression as x approaches infinity.
lim(x->∞) GMm/x = 0
Therefore, the limit of the work W2 as the rocket rises infinitely far from the Earth is zero.
W2 = 0 (mile-pounds)