A car driver going at some speed v suddenly finds a wide wall at a distance r.Should he apply brakes or turn the car in a circle of radius r to avoid hitting the wall?

Let F₁ be the braking force, so as to stop the car just in the distance r. 

Then,

work done by the braking force = kinetic energy of the car

or F₁ x r = 1/2 mv²

or F₁ = (mv²) / 2r                                 ...1

Let F₂ be th force required (centripetal force force) for taking a circular turn of raduis r.

Then,

F₂ = mv² / r                                         ...2

From the equations (1) and (2), we have

F₁ = 1/2 F₂

Since the force needed to apply brakes is just half of the force required to take the circular turn, it is better to apply brakes.