Problem 31E

The kinetic energy KE of an object of mass m moving with velocity v is defined as KE=12mv2 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:12mv2212mv12, 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=x1x2f(x)dx=t1t2f(s(t))v(t)dt Then use Newtons Second Law of Motion (force=massacceleration) 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 20mi/h? Note: Divide the weight in pounds by 32ft/s2, the acceleration due to gravity, to find the mass, measured in slugs.

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