Find the final speed using a work integral with F(x)=3sin^2(pi*x).

Given a variable force function F(x)=3*sin^(pi*x), we compute the final speed of a particle by computing a work integral and using the work-energy theorem.

This integral is particularly challenging, because the integral of sin^2(pi*x) requires the use of a power reducing trig identity sin^2(theta)=1/2*(1-cos(2*theta)). After applying the identity, we quickly guess the antiderivative of each term and we find the total work done on the particle by integrating the force function.

Finally, we apply the work-energy theorem W=delta(K). With K_i=0 because the particle started at rest, we write K_f as 1/2mv_f^2 and set equal to the total work done on the particle. Solving for v_f and plugging in the numbers, we arrive at the final speed of the particle.

[slayeriv]

Very useful, I am surprised I haven’t found this channel sooner😊