Work in a gravitational field For large distances from the surface of Earth, the gravitational force is given by F(x) = GMm / (x+R)², where G = 6.7×10^−11 N m²/kg² is the gravitational constant, M = 6×10^24 kg is the mass of Earth, m is the mass of the object in the gravitational field, R = 6.378×10⁶ m is the radius of Earth, and x≥0 is the distance above the surface of Earth (in meters).


a. How much work is required to launch a rocket with a mass of 500 kg in a vertical flight path to a height of 2500 km (from Earth’s surface)?

Verified step by step guidance

1. Identify the force function given by the problem: \(F(x) = \frac{GMm}{(x+R)^2}\), where \(x\) is the height above Earth's surface, \(G\) is the gravitational constant, \(M\) is Earth's mass, \(m\) is the rocket's mass, and \(R\) is Earth's radius.
2. Recognize that work done against a variable force over a distance is calculated by integrating the force over that distance. Since the force depends on \(x\), set up the integral for work as \(W = \int_{0}^{h} F(x) \, dx\), where \(h\) is the height to which the rocket is launched (converted to meters).
3. Substitute the expression for \(F(x)\) into the integral: \(W = \int_{0}^{h} \frac{GMm}{(x+R)^2} \, dx\).
4. Evaluate the integral by recognizing that \(\int \frac{1}{(x+R)^2} \, dx = -\frac{1}{x+R} + C\). Apply the limits of integration from \$0\( to \)h\( to find \)W = GMm \left( \frac{1}{R} - \frac{1}{R+h} \right)$.
5. Plug in the known values for \(G\), \(M\), \(m\), \(R\), and \(h\) (remembering to convert \(h = 2500\) km to meters) into the expression for \(W\) to find the work required to launch the rocket to the specified height.

Key Concepts

Gravitational Force and Potential Energy

The gravitational force between two masses decreases with the square of the distance between their centers. The potential energy associated with this force depends on the position relative to Earth’s center, and work done against gravity changes the potential energy of the object.

Work Done by a Variable Force

When force varies with position, work is calculated as the integral of the force over the displacement. For gravitational force that depends on distance, the work to move an object from one height to another is found by integrating the force function with respect to distance.

Integration in Calculus

Integration is used to find the total accumulation of quantities, such as work done by a variable force. In this problem, integrating the gravitational force function from the initial to final position yields the total work required to move the rocket to the desired height.