Motion in a Straight Line

Given, initial speed of engine =u
Speed of engine last compartment =v
Let the length of train be I and the speed of mid-point of train be v′.
Using third equation of motion,
v2=u2+2as
⇒‌‌v′2=u2+2al∕2
⇒‌‌v′2−u2=al . . . (i)
‌ Also, ‌‌‌v2−u2=2al⇒a=‌

v2−u2

2l

Substituting the value of a in Eq. (i), we get
v′2−u2‌‌=‌

v2−u2

2l

⋅l=‌

v2−u2

2

⇒‌‌v′‌‌=√‌

v2−u2

2

+u2=√‌

v2+u2

2