Why does a car take longer to stop is brakes are applied too hard?
When you try to push a broken-down car with its handbrake still on, it does not move. Explain other forces that are acting on the car to produce a net force of zero.
When you hit the brakes, the friction from the brakes slows down the car wheels, which slows down the car from the friction generated between the car wheels and the road surface. In normal car brake operation, the wheels never actually stop rotating until the car comes to a stop. This means that the friction between the car and the road surface is actually static friction (which is the friction between two non-moving surfaces). When you slam on the brakes too hard, the friction from the brakes can overcome the static friction between the wheels and the road surface, which causes the wheels to lock up (stop spinning), and instead start skidding. This skidding creates instead kinetic or dynamic friction (which is the friction between surfaces in relative motion to one another). In general, dynamic friction is lower than static friction (which you can see if you put a block on an angled ramp. At some angles, the block will remain stationary, but if you just push it, the block slides down to the bottom.) so when the car is skidding, there is less friction and hence less deceleration than when the car is braking normally. Thus the car takes longer to stop if the brakes are applied too hard.
This is the entire reason that all cars now come with ABS (anti-lock braking systems), which detect when the wheels are starting to skid and reduces brake pressure accordingly to make sure you don't enter a skid. Before ABS was introduced, people were taught to pump the brakes on and off for hard braking (but naturally in those situations, they'd forget due to stress/panic). Interestingly, ABS equipped cars perform worse on gravel and other loose road surfaces than cars without it. Don't remember why though.
The car with the hand-brake on seems pretty self explanatory to me. You are pushing a car, so you are exerting a force (say, forwards) on the car. The friction between the wheels and the road created by your pushing creates a torque on the wheels, which is resisted by the friction between the brake pads and the wheel (because the hand-brake is on). This friction is presumably greater than the force that you can exert to push on the car, so the net force is zero.
It's not impossible to push a car with the handbrake on though. If you had enough people (say, ten 17 year olds with too much time on their hands), you can get enough force to overcome the friction from the brakes and thus move the car forwards.
Hope that helps (and hope I'm not wrong).
