A car driver has to make an emergency stop. The braking distance depends on the speed of the car. For the same braking force, what happens to the braking distance if the speed doubles?

For a situation like this it is best to think of the energy of the car!The car has a kinetic energy = 1/2mv² so if the speed has doubled the new kinetic energy of the car will be 1/2m(2v)² meaning the kinetic energy has increased by a factor of 4. To top the car work must be done by the brakes. The work done will also have gone up by a factor of 4. As work = f x d and the force is the same, the braking distance will have increased by a factor of 4.