Question

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.

Answer 1

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(v²+1)dv = 8dx
Integrating both sides
∫ v(v²+1)dv = ∫ 8dx
`(v^2)/(4)+(v^2)/(2)=8x+c`……….(1)
Multiplying by 4 on both sides
V⁴ + 2v2 = 32x + 4c
Substituting v=0 and x=0 in (1)
c=0 From (1)
V⁴ + 2v² = 32x ………….(2)
Case 1 : x=20 m
V⁴ + 2v² = 32 x 20 ………(From 2)
V⁴ + 2v² – 640 = 0
Solving the equation
V² = 24.3180
V=4.9361 m/s
18⁴ + 2 x 18² = 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