Chapter 5.4, Problem 31E

The kinetic energy KE of an object of mass m moving with velocity v is defined as $\text{KE}=\frac{1}{2}m{v}^2$. If a force f(x) acts on the object, moving it along the x-axis from x₁ to x₂, the Work-Energy Theorem states that the net work done is equal to the change in kinetic energy: $\frac{1}{2}m{v}_2^2-\frac{1}{2}m{v}_1^2$, where v₁ is the velocity at x₁ and v₂ is the velocity at x₂.

(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 (4.5.5) to show that

$W={\displaystyle {\int }_{x_1}^{x_2}f\left(x\right)dx=}{\displaystyle {\int }_{t_1}^{t_2}f\left(s\left(t\right)\right)v\left(t\right)dt}$

Then use Newton’s Second Law of Motion (force = mass × 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/s², the acceleration due to gravity, to find the mass, measured in slugs.)