A body starts from origin with initial velocity U and retardation KV^3 where V is instantaneous velocity...?

Question

A body starts from origin with initial velocity U and retardation KV^3 where V is instantaneous velocity of the body.

A) velocity of the body at any instant is √(2U^2Kt + 1)

B) velocity of the body at any instant is U/[√(2U^2Kt + 1)]

C) the x coordinates of the body at any instant is 1/UK * [√(2U^2Kt + 1) - 1]

D) Magnitude of initial acceleration of the body is KU^3

More than one options may be correct!

Answer 1

I eat your face.

Answer 2

given > retardation (a) = - KV^3

a  = dV/dt = - KV^3

dV/V^3 = - K dt

V^(- 3) dV = - K dt

integrating

- 1/(2V^2) = - K*t + C

at t=0, V = U

- 1/(2U^2) = C

- 1/(2V^2) = - K*t - 1/(2U^2)

1/V^2 = 2K*t + 1/U^2 = [1 + 2 K t U^2] / U^2

upside down

V = U /√[2U^2 K t + 1]

option (B) is correct

-----------------------------------

V = U /√[2U^2 K t + 1]

dx/dt = V = U /√[2U^2 K t + 1]

dx = U * dt /√[2U^2 K t + 1]

integrate

===============

[2U^2 K t + 1] = p^2

dt = pdp/U^2 K

integral { [U* pdp/U^2K] [1/p]

integral { [dp/UK] = [1/UK] p = [1/UK] *√[2U^2 K t + 1]

==========

x =  [1/UK] *√[2U^2 K t + 1] + C1

at t=0, x=0

C1 = [1/UK]

x =  [1/UK] *{√[2U^2 K t + 1] + 1}

rest you may check yourself