#### Answer

(a) At point A, the minimum speed is 7.7 m/s
(b) At point B, the minimum speed is 10 m/s

#### Work Step by Step

(a) At the midpoint between point A and point B, the potential energy is at a maximum of 5.0 J. For the particle at point A to reach point B, the total energy must be at least 5.0 J. We can find the minimum kinetic energy required at point A.
$K+U = E$
$K = E-U$
$K = 5.0~J - 2.0~J$
$K = 3.0~J$
We can find the minimum speed of the particle at point A.
$K = 3.0~J$
$\frac{1}{2}mv^2 = 3.0~J$
$v^2 = \frac{(2)(3.0~J)}{m}$
$v = \sqrt{\frac{(2)(3.0~J)}{m}}$
$v = \sqrt{\frac{(2)(3.0~J)}{0.10~kg}}$
$v = 7.7~m/s$
At point A, the minimum speed is 7.7 m/s.
(b) At the midpoint between point A and point B, the potential energy is at a maximum of 5.0 J. For the particle at point B to reach point A, the total energy must be at least 5.0 J. We can find the minimum kinetic energy required at point B.
$K+U = E$
$K = E-U$
$K = 5.0~J - 0$
$K = 5.0~J$
We can find the minimum speed of the particle at point B.
$K = 5.0~J$
$\frac{1}{2}mv^2 = 5.0~J$
$v^2 = \frac{(2)(5.0~J)}{m}$
$v = \sqrt{\frac{(2)(5.0~J)}{m}}$
$v = \sqrt{\frac{(2)(5.0~J)}{0.10~kg}}$
$v = 10~m/s$
At point B, the minimum speed is 10 m/s.