Problem 26 - Calculus Concepts

Question

Rocket Propulsion The work required to propel a 10 -ton rocket an unlimited distance from the surface of Earth into space is defined in terms of force and is given by the improper integral $$ w(x)=\int_{4000}^{\infty} \frac{160,000,000}{x^{2}} d x $$ The expression \(\frac{160,000,000}{x^{2}}\) is force in tons. The variable \(x\) is the distance, measured in miles, between the rocket and the center of Earth. a. What are the units of work in this context? b. Calculate the work necessary to propel this rocket infinitely into space.

Step-by-Step Solution

Answer

a. The units of work are ton-miles. b. The required work is 40,000 ton-miles.

Step 1: Identify Units of Work

In physics, work is calculated as the product of force and distance. Given that the force is in tons, which can be converted to pounds (1 ton = 2000 pounds), and the distance is in miles, the units of work in this context would be in ton-miles. This is because the integral will result in the product of force over a distance.

Step 2: Setup the Integral for Calculation

The work required is given by the improper integral \( \int_{4000}^{\infty} \frac{160,000,000}{x^{2}} \, dx \). This integral calculates the total work done from a distance of 4000 miles (the approximate radius of the Earth) to infinity.

Step 3: Evaluate the Improper Integral

The integral \( \int \frac{1}{x^2} \, dx \) has an antiderivative of \( -\frac{1}{x} \). Applying this to our integral:\[ w(x) = \left[-\frac{160,000,000}{x}\right]_{4000}^{\infty} \] This evaluates to:\[ \lim_{b \to \infty} \left(-\frac{160,000,000}{b} + \frac{160,000,000}{4000} \right) \] Since \( \frac{160,000,000}{b} \to 0 \) as \( b \to \infty \), the integral simplifies to:\[ w(x) = 0 + \frac{160,000,000}{4000} \]

Step 4: Simplify the Result

Plug in the value 4000:\[ w(x) = \frac{160,000,000}{4000} = 40,000 \text{ ton-miles}\].This is the work required to move the rocket from the surface of the Earth to an infinite distance away.

Key Concepts

Improper Integral

An improper integral is a type of integral where the integrand is undefined or the limits of integration extend to infinity. In the context of rocket propulsion, improper integrals are useful because the work needed to propel an object into space cannot be quantified with finite bounds. In our exercise, the integral starts at 4000 (the Earth's approximate radius in miles) and extends to infinity, representing the concept of an object traveling indefinitely into space.
Understanding improper integrals involves recognizing that they often require special techniques to evaluate, because as the limits extend to infinity, typical arithmetic operations might not apply straightforwardly. We approach this by considering the limit of the integral, which helps in handling the infinite boundary effectively.

Work in Physics

In physics, 'work' is described as the amount of energy transferred by a force moving an object over a distance. It serves as a critical concept when analyzing forces and motions. The unit of work is often expressed in terms such as 'joules' in SI units, but in our exercise, it is calculated as 'ton-miles.'

• Work is calculated by multiplying force by distance.
• In the rocket propulsion problem, force is given in tons and converted to ton-miles for the work calculation based on the distance the rocket travels.

This understanding of work allows us to determine the energy required to propel the rocket from Earth into the expanse of space.

Force and Distance

Force and distance are crucial elements to consider when calculating work. In our rocket propulsion scenario, the force involved is represented by the function \(\frac{160,000,000}{x^{2}}\). This shows how the force diminishes with increasing distance from the Earth.

• Force is the interaction that causes an object to move or accelerate.
• Distance describes how far the object has traveled under the force’s influence.

Successfully calculating work involves understanding the relationship between force and distance, as the two factors directly influence the energy expended in moving the object. This is expressed by integrating the force function over distance, acknowledging their intertwined nature.

Antiderivative

An antiderivative is a function that reverses the process of differentiation, allowing us to integrate and find the area under a curve. For the problem at hand, finding the antiderivative is crucial in evaluating the improper integral.
For the force function \(\frac{1}{x^2}\), the antiderivative is \(-\frac{1}{x}\). This result is essential as it transforms the integral into an expression that can be evaluated between the given limits, from the Earth's radius of 4000 miles to infinity.
Understanding how to find antiderivatives enables us to solve a wide array of physics problems where integration is necessary, providing a way to transition from differential forms to cumulative or total values.

Improper Integral Evaluation

Evaluating an improper integral involves calculating the limit of the integral as it approaches infinity. This method allows us to resolve the problem of infinite bounds in integrals. In our rocket propulsion example, the evaluation process includes:

• Identifying the antiderivative \(-\frac{160,000,000}{x}\).
• Substituting the bounds: \(\left[-\frac{160,000,000}{x}\right]_{4000}^{\infty}\).
• Understanding that as \(x\) approaches infinity, \(-\frac{160,000,000}{x}\) approaches 0.

Through these steps, the final calculation evaluates the work done to ensure the rocket reaches infinite space, yielding a result of 40,000 ton-miles. It represents how much energy is needed, demonstrating the practical application of improper integrals in physics.