Why are trains so hard to stop? Since there are brakes on every car, shouldn't the stopping time of the entire train be the same as the stopping time of a single car?
Is this because the cars themselves take a long time to stop, or because train braking is less efficient than typical braking, or another reason?
 A: Suppose you have a vehicle with a mass $m$ moving at some speed $v$ and the friction coefficient between the wheels and the road is $\mu$. The maximum deceleration is then:
$$ a_\text{max} = \frac{F}{m} = \frac{mg\mu}{m} = g\mu $$
The time taken to stop will be:
$$ t = \frac{v}{a} = \frac{v}{g\mu} $$
And the distance taken to stop will be:
$$ s = \frac{v^2}{2a} = \frac{v^2}{2g\mu} $$
So the stopping time and distance are inversely proportional to the friction coefficient $\mu$.
The maximum braking force is determined by the friction coefficient between the wheels and the surface. In the case of cars even a basic family car will have a tyre-road friction coefficient of around 0.75, and for a sports card it will be nearer 1.
For trains the wheels and the rail are both steel, and the steel-steel friction coefficient is around 0.25. So the stopping time and distance will, at best, be three to four times greater than a car. In practice locking the wheels of a train causes damage to the wheels and rail that is very expensive to fix, so train brakes are designed to provide only about 75% of the maximum possible braking effort.
Finally we should note that a train under braking is a potentially unstable object. If the braking efficiency varies along the train it could concertina and derail. For this reason long trains will not use the maximum braking possible unless there's a very good reason.
