Problem 31 - Calculus: Early Transcendentals

Question

The kinetic energy \( KE \) of an object of mass m moving with velocity \( v \) is defined as \( KE = \frac{1}{2} mv^2 \). If a force \( f(x) \) acts on the object, moving it along the x-axis from \( x_1 \) to \( x_2 \), the Work-Energy Theorem states that the net work done is equal to the change in kinetic energy: \( \frac{1}{2} mv^2_2 - \frac{1}{2} mv^2_1\) , where \( v_1 \) is the velocity at \( x_1 \) and \( v_2 \) is the velocity at \( x_2 \). (a) Let \( x = s(t) \) be the position function of the object at time \( t \) and \( v(t) \), \( a(t) \) the velocity and acceleration functions. Prove the Work- Energy Theorem by first using the Substitution Rule for Definite Integrals (5.5.6) to show that $$ W = \int_{x_1}^{x_2} f(x) dx = \int_{t_1}^{t_2} f(s(t)) v(t) dt $$ The use Newton's Second Law of Motion (force = mass \( \times \) acceleration) and the substitution \( u = v(t) \) to evaluate the integral. (b) How much work (in ft-lb) is required to hurl a 12-lb bowling ball at 20 mi/h? (Note: Divide the weight in pounds by \( 32 ft/s^2 \), the acceleration due to gravity, to find the mass, measure in slugs.)

Step-by-Step Solution

Answer

The work required is approximately 161.3 ft-lb.

Step 1: Establish Relationship with Substitution Rule

We are given that the work done \( W \) can be expressed as \( W = \int_{x_1}^{x_2} f(x) \, dx \). Using the position function \( x = s(t) \), the substitution rule allows us to rewrite the integral with respect to time: \( x = s(t) \), \( dx = v(t) \, dt \). So \( W = \int_{t_1}^{t_2} f(s(t)) v(t) \, dt \).

Step 2: Apply Newton's Second Law of Motion

Newton's Second Law states \( f(x) = ma(t) \), where \( a(t) = \frac{dv}{dt} \). Substituting \( a(t) = \frac{dv}{dt} \) into the integral, we have: \( W = \int_{t_1}^{t_2} m \frac{dv}{dt} v(t) \, dt \).

Step 3: Evaluate the Integral Using Substitution u = v(t)

Let \( u = v(t) \) so that \( du = \frac{dv}{dt} \, dt \). This transforms the work integral to: \( W = \int_{v_1}^{v_2} mu \, du \).

Step 4: Integrate and Solve

The integral becomes: \( W = m \int_{v_1}^{v_2} u \, du = m \left[ \frac{1}{2}u^2 \right]_{v_1}^{v_2} = m \left( \frac{1}{2}v_2^2 - \frac{1}{2}v_1^2 \right) \). This shows the net work done equals the change in kinetic energy, proving the Work-Energy Theorem.

Step 5: Calculate Work for Bowling Ball

Convert the weight to mass in slugs: \( \text{mass} = \frac{12 \text{ lb}}{32 \text{ ft/s}^2} = 0.375 \text{ slugs} \). Convert velocity from mi/h to ft/s: \( 20 \text{ mi/h} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = 29.333 \text{ ft/s} \). Calculate change in kinetic energy: \( KE = \frac{1}{2} \times 0.375 \text{ slugs} \times (29.333^2 - 0^2) \text{ ft}^2/\text{s}^2 \approx 161.3 \text{ ft-lb} \).

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object has due to its motion. It is a fundamental concept in physics, often represented by the formula: \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. This formula shows that kinetic energy depends on both the mass and the square of the velocity of the object. This means that even a slight increase in velocity can lead to a significant increase in kinetic energy. When an object accelerates, its velocity changes, and so does its kinetic energy. In the context of the Work-Energy Theorem, this change in kinetic energy is directly related to the net work done on the object. Understanding kinetic energy is essential for analyzing motion and predicting how an object will behave under different forces.

Definite Integral

The definite integral is a key concept in calculus that helps us calculate quantities such as areas under curves or, in this case, work done by a force over a distance. For a function \( f(x) \), the definite integral from \( x_1 \) to \( x_2 \) is represented as \( \int_{x_1}^{x_2} f(x) \, dx \). In physics, this integral can be used to calculate physical properties like work. The integral tells us the total accumulation of a quantity, adding up all the small parts over an interval. In terms of work, it involves summing the infinitesimal contributions of force acting along the path from \( x_1 \) to \( x_2 \). By understanding how definite integrals function, we can grasp how forces act over distances to accomplish work.

Newton's Second Law

Newton's Second Law of Motion is a pivotal principle in physics, expressed as \( F = ma \). This law states that the force acting on an object is equal to the mass \( m \) of the object multiplied by its acceleration \( a \). It connects the concepts of force, mass, and motion, providing a framework for understanding how objects move when subjected to forces.In the Work-Energy Theorem, Newton's Second Law is instrumental in transforming the integral expression for work into a form that makes the change in kinetic energy evident. By substituting \( f(x) = ma \), where acceleration is the derivative of velocity \( a(t) = \frac{dv}{dt} \), we can link force directly to the velocity change of the object. This connection is crucial for proving the relationship between work and kinetic energy.

Substitution Rule for Definite Integrals

The Substitution Rule is a powerful tool in calculus, allowing us to simplify complex integrals by changing variables. In the context of proving the Work-Energy Theorem, we utilize this rule to convert the integral with respect to displacement \( x \) into one with respect to time \( t \).When the position of an object is given by \( x = s(t) \), and we know the velocity \( v(t) \) and acceleration \( a(t) \), we can make such a substitution. By setting \( dx = v(t) \, dt \), it allows us to express the integral for work as \( \int_{t_1}^{t_2} f(s(t)) v(t) \, dt \). Further simplification with the substitution \( u = v(t) \) continues to the final integral form. This substitution method makes it easier to evaluate the integral and observe the work-energy relationship clearly.