Can friction be static and kinetic at the same time? We all know the classical question about a car driving at constant speed on a circular banked highway with friction.
Turns out that there is a range of allowed velocities that the car can drive in order to maintain a circular trajectory.
If the speed is above some upper limit, the car will slip upward (centrifugal force is stronger than gravity and friction).
If the speed is below the lower limit, the car will slip downward (gravity is stronger than centrifugal force and friction).
All the velocities between these values will result in the same circular trajectory.
So far so good.
Now lets assume that the car is driving at the upper limit of the velocity and does not slip. So the static friction is pointed downward, perpendicular to the car trajectory.
Now the driver hits the breaks, and lets assume that the wheels immediately stop rolling and locks into a fixed position without rotation.
What happens to the friction just after the wheels are locked?
The velocity decreases but it is still within the allowed values, hence the static friction is still pointed downward and prevent the car from slipping upward, so the car maintains its circular trajectory.
But now the wheels are locked, so there is also kinetic friction between the wheels and the highway in the direction parallel to the car trajectory, and opposite to its motion.
So what is going on here?
can the friction be kinetic in one direction (parallel to the trajectory and backwards), while at the same time be static in the other direction (perpendicular to the trajectory and downward)?
Note that the lower limit of the velocity is unimportant to this question. In fact, you can decrease the angle of the highway so that the car will not slip downward even when it is stationary, so the lower limit of the allowed velocity can be zero.
In that case, once the wheels are locked, the car velocity will decrease until the car reaches a full stop, while maintaining the initial circular trajectory during that whole time.
So what's up with the friction here?        
 A: Well the answer to your question about mixed static and kinetic is yes you can have both static and kinetic friction between two objects at the same time (particularly easily with tires). However, it doesn't happen in the way you're describing.
First off, when two objects interact there is always one net force between the two objects. (Or you could call it one force and it's equal and opposite reaction force). This force vector is usually split up into a "normal", or "support" force and a frictional force. If there is any slipping or sliding the direction of the frictional force is always* in the direction that helps stop the sliding.
First lets taking the slope out of the curve so it is just a flat turn and then we can add the banking back in in the next section.
Locking your wheels in a turn.
When you lock your wheels, suddenly the tire surface is moving forward while the road surface is still stationary. You lose all static friction and the direction of the friction points directly backward (opposing the motion). Assuming the car is equally balanced the entire car will slide in the direction it was going drifting to a stop (it will also continue spinning as the same speed). All steering control is lost. This is half of the reason for the anti lock brake systems in cars, once the wheel isn't rotating, the direction it is pointing is basically irrelevant.
Locking your wheels in a banked turn.
This scenario is similar, the car will still drift in the direction it was going, so it will rise outward until the velocity is low enough that the gravity would pull it back down the curve. Again with the wheels locked there will only be kinetic friction until the car comes to rest.
How can you have mixed friction?
So then the question is when is there ever both static and kinetic friction. The answer may surprise you that the vast majority of the time your tires are experiencing both. This is because the tires are very flexible and stretchy. This flexibility allows part of the contact patch of the tire to be slipping against the road surface while the rest of the contact patch is sticking statically to the road. The net force is the sum of both forces. In this case they are still always pointing in the direction that would prevent slipping. For more information you can google "slip angle" for explanations on how stick and slipping work together to provide cornering force.
