An engine of a train, moving with uniform acceleration, passes the signal-post with velocity u and the last compartment with velocity v. The velocity with which middle point of the train passes the signal post is :

CONCEPT:

According to the third equation of the motion, we have;

v2 - u2 = 2as

Here we have v as the final velocity, u is the initial velocity and s is the length or distance.

CALCULATION:

Let the acceleration of the train is " a "

and the length of the train is " l "

The initial velocity is " u "

and final velocity is  " v "

The third equation of motion, it is written as;

v2 - u2 = 2al    -----(1)

When the train is in the middle point, the length of the train becomes , therefore third equation of the motion becomes,

v'2 - u2 = 2 a 

v'2 - u2 = al     ----(2)

Now, by solving equations (1) and (2) we have;

v2 - u2 = 2al 

⇒ l = 

On putting this value in equation (2) we have;

v'2 - u2 =  a

⇒ v'2 - u2 =  

⇒ v'2 = 

⇒ v' = 

Hence, option 3) is the correct answer.