Which Method is better to avoid Road Accident? If I am driving at night and fall asleep, and when I wake up I see a wall in front of me.
What should my instincts be :

*

*Should I just apply brakes?


*Or should I apply brakes and try to turn the car in a circular curvature?


*Any other method to save my life and my car?
Any help is greatly appreciated.
 A: It depends on many factors, such as the distance to the wall, the speed at the moment you wake up, the type of tires, ...
The energy of the car is $1/2 mv^2$, so you want to dissipate as much velocity in the forward direction as possible. The most efficient way to do this is to slam the brake. Cons: the ABS system (which most cars are equipped with nowadays) makes the braking distance much longer, so probably you won't avoid the collision. Turning the vehicle will change the speed from v to $v*cos(\theta)$ where $\theta$ is the angle you manage to turn before hitting the wall. To decrease the forward speed by half you would need to turn $60^0 $. I only have data about how safe it is to turn by $90^0$, and it indicates that for an average car the maximum speed for a safe $ 90^0$ turn is about $58 km/h 
 $$(36mph)$. If you go faster than that, the car will probably overturn.
So in short, I think the best chances are if you slam the brake and when the speed has decreased below $ 58 km/h$ you try to turn the car.
If you are about to hit an oncoming vehicle instead of a wall, the game changes completely. In that case the best option is to turn your car to try to avoid collision at all cost. Your chances of survival are better if you fall off a cliff than if you hit another car head to head.
A: The fundamental rule (answer) is: do everything you can to minimize the magnitude of velocity perpendicular to the wall.
After that, it's an engineering question.  Basically, at what g-force will your tires break traction when applying brakes and moving straight forward?  At what g-force will your tires break traction when making a turn of radius R at your current speed (magnitude of velocity)?   And, finally, as PhilipW points out, the longer you can make your path before hitting the wall, the more time you have to decelerate.
I strongly suspect that the tires have a different effective coefficient of (static, not rolling) friction in the X and Y axes of the wheel coordinates (along the rolling direction and cross-tread) , which means figuring out what path you can follow to maximize the integral of  max_deceleration over said path is not at all trivial.
For that matter, executing a deliberate break-of-traction and a turn of $\pi$ (180 degrees) and flooring it might work well too, with the added advantage that crashing "backwards" greatly reduces the stress on your body due to support from the seatback&headrest.   Admittedly this is not a maneuver most of us could execute correctly even wide awake.
A: What follows are some observations on your question. They must not be used by themselves to guide your driving decisions.
Suppose that you find yourself a distance b from a wall and moving straight towards it at speed u. We shall ignore your reaction time.
In order to stop in a straight line without hitting the wall, your acceleration in a direction away from the wall (deceleration, if you like) needs to be $\tfrac 12 \frac {u^2}{b}$. [This was obtained from $v^2=u^2 + 2as$, which assumes constant acceleration.]
Suppose that, instead, you maintain your speed and manage to turn in a quarter-circle of radius b, that is just avoiding scraping the wall. In that case the acceleration will have to be $\frac{u^2}{b}$ directed towards the centre of the circle. [A hand-waving style reason for its being greater than the straight line deceleration is that not only do you get rid of the car's velocity component at right angles to the wall, but you give it a new velocity component parallel to the wall.]
Any acceleration of the car is due to frictional forces on the tyres from the road. There is a limit to the magnitude of frictional force that can be supplied. So it looks as if there's less chance of stopping without skidding (or of abs kicking in) if you go in the curved path. What's more, in a curved path, the acceleration is at an angle to the 'forward' direction of the tyre, which is not what tyres are ideally suited to.
What needs looking into now is the applying brakes at the same time as steering an avoidance course, but I doubt if it's the best strategy.
