Answer
(a) $v = 2.39~m/s$
(b) The magnitude of the acceleration is $9.42~m/s^2$ and the acceleration is directed away from the wall.
Work Step by Step
(a) The kinetic energy of the block will be equal to the sum of the work done by the force, the friction, and the spring.
$\frac{1}{2}mv^2 = F~x - mg~\mu_k~x - \frac{1}{2}kx^2$
$v^2 = \frac{2F~x - 2mg~\mu_k~x - kx^2}{m}$
$v = \sqrt{\frac{2F~x - 2mg~\mu_k~x - kx^2}{m}}$
$v = \sqrt{\frac{(2)(82.0~N)(0.800~m) - (2)(4.00~kg)(9.80~m/s^2)(0.400)(0.800~m) - (130.0~N/m)(0.800~m)^2}{4.00~kg}}$
$v = 2.39~m/s$
(b) We can use a force equation to find the acceleration.
$\sum F = ma$
$F-mg~\mu_k-kx = ma$
$a = \frac{F-mg~\mu_k-kx}{m}$
$a = \frac{82.0~N-(4.00~kg)(9.80~m/s^2)(0.400)-(130.0~N/m)(0.800~m)}{4.00~kg}$
$a = -9.42~m/s^2$
The magnitude of the acceleration is $9.42~m/s^2$ and the negative sign means that the acceleration is directed away from the wall.
