An express train without its rake can go 24 km an hour, and the speed is diminished by a quantity that varies as the square root of the number wagons attached .If it is known that it is four wagons its speed is 20 km/hthe greatest number of wagons with which the engine can just move is

Speed of train without wagons = 24 km/h

Reduction in speed with 'n' wagons $$\propto \sqrt{n}$$

Reduction in speed with 'n' wagons $$= k\sqrt{n}$$

Speed of train with 4 wagons = 20 = 24 $$-k\sqrt{4}$$

2k = 4

k = 2

Now, 

Consider the case when the speed of train is zero. 

$$24 - 2\sqrt{n} = 0$$

$$24 = 2\sqrt{n}$$

$$12 = \sqrt{n}$$

$$n = 12^2$$

n = 144

Hence, total number of wagons will be 143 when the train cna just move.