Answer
The mug does not slide.
Work Step by Step
We can find the maximum possible magnitude of acceleration of the mug which can be provided by the force of static friction.
$F_f = ma$
$mg~\mu_s = ma$
$a = g~\mu_s$
$a = (9.80~m/s^2)(0.50)$
$a = 4.90~m/s^2$
We can find the deceleration of the car during the braking period.
$a = \frac{v^2-v_0^2}{2x}$
$a = \frac{0-(20~m/s)^2}{(2)(50~m)}$
$a = -4.0~m/s^2$
Since the mug's maximum possible magnitude of acceleration provided by the force of static friction is greater than the magnitude of deceleration of the car, the mug does not slide.
