Answer
The speed of the box after it has traveled 14.0 meters is 8.16 m/s.
Work Step by Step
We can find the work done by the force between x = 0 and x = 14.0 meters.
$W = \int_{0}^{14.0}F_x~dx$
$W = \int_{0}^{14.0}(18.0~N)-(0.530~N/m)~x~dx$
$W = (18.0~N)~x-(0.265~N/m)~x^2 ~\vert_0^{14.0}$
$W = (18.0~N)(14.0~m)-(0.265~N/m)(14.0~m)^2$
$W = 200~J$
We can use the work done by the force to find the speed of the box.
$\frac{1}{2}mv^2= 200~J$
$v^2 = \frac{400~J}{m}$
$v = \sqrt{\frac{400~J}{6.00~kg}}$
$v = 8.16~m/s$
The speed of the box after it has traveled 14.0 meters is 8.16 m/s.
