The acceleration of the train starting from rest at any instant is given by the expression `a=8/(v^2+1)` where v is the velocity of train in m/s. Find the velocity of the train when its displacement is 20 m and its displacement when velocity is 64.8 kmph.

#### Solution

Given : `a=8/(v^2+1)`

To find : Velocity when displacement is 20 m

Displacement when velocity is 64.8 kmph.

Solution :

`a=v(dv)/(dx)`

`v(dv)/(dx)=8/(v^2+1)`

v(v^{2}+1)dv = 8dx

Integrating both sides

∫ v(v^{2}+1)dv = ∫ 8dx

`(v^2)/(4)+(v^2)/(2)=8x+c`……….(1)

Multiplying by 4 on both sides

V^{4} + 2v2 = 32x + 4c

Substituting v=0 and x=0 in (1)

c=0 From (1)

V^{4} + 2v^{2} = 32x ………….(2)

Case 1 : x=20 m

V^{4} + 2v^{2} = 32 x 20 ………(From 2)

V^{4} + 2v^{2} – 640 = 0

Solving the equation

V^{2} = 24.3180

V=4.9361 m/s

18^{4} + 2 x 18^{2} = 32x …………..(From 2)

1.5624 = 32x

x = 3300.75 m**When displacement of train is 20 m,then velocity is 4.9361 m/s****When velocity of the train is 64.8 kmph,then its displacement is 3300.75m**