A man starts rowing his stationary cuboidal boat of base area `A=10m^(2)`. The driving force on the boat due to rowing is `100N` in the direction of motion. Find the maximum velocity that the boat can achieve. Also find the time in which he will attain half of this maximum velocity. `[` Take coefficient of viscosity of water `=15` poise `]` The depth of the lake is `10m` and the combined mass of man and the boat to be `150kg. (u=0`, velocity gradient uniform)

Answer

`(200)/(3)m//s,t=100ln2 ` seconds

Explanation

At equilibrium
`F_(ext)=F_(v)`
`F_(ext)=100N` and `F_(v)=etaA(V_(max))/(D)=1.5xx10xx(V_(max))/(D)`
`:. 100 =1.5V_(max)`
`rArr V_(max)=(200)/(3)m//s`
At any instant
`F_("ext")-F_(v)=m(dv)/(dt)`

`=100=etaA(v)/(D)=m(dv)/(dt).`
`:. 100-1.5v=150(dv)/(dt)`
from `(1)underset(0)overset(t)int dt=underset(0)overset((V_(max))/(2))int150(dv)/(100-1.5v)`
`rArrt=150underset(0)overset(100//3)int(ln(100-1.5v))/(-1.5)`
`rArrt=100ln2` seconds