IIT JEE , AIEEE Forum - Physics , Mechanics

kislay

11. The current velocity of river grows in proportion to the distance from its bank and reaches the maximum value v₀ in the

middle. near the bank the velocity is zero. A boat is moving along the bank in such a manner that it is always perpendicular

to the current. the speed of the boat in still water is u. find the distance through which the boat crossing the river will

be carried away by the current if the width of the river is c. Also determine the trajectory of the boat.

ans
(a)cv₀/2u
(b)y²=ucx/v₀

anubhav_89

Taking vertical dir. time taken to cross river=c/u. Let the middle of river be x axis, so the expression for velocity becomes v=v0-Kx, where k is const. Putting x=c/2, i.e. on the banks v=0,
so v0=kc/2 or k=2v0/c Now
v0=v+2v0x/c where x is dist from middle of river
v=u+at Comparing
at=2v0x/c
a=2v0x/ct

for middle of river
v0^2=2as
v0^2=2v0xs/ct
v0^2=2*2v0*(c/2)*s / (c*(c/2u))
s=v0c/4u

so total displacement while crossing river =2s
=v0c/2u

kislay

hey anubhav u have done a gr8 job, but by that method i am not able to find the trajectory of the boat.
pls help.

deepak_agarwal

hey anubhav u have seriously produce a nic ans..the problem was challenging one

kislay

hey experts pls help in finding trajectory of the boat

sudeep.kumar

Dear Kislay,
For the trajectory,

y = ut    (as the boat always travels perpendicular to flow, v_y = u)
so, dy = udt

v_x = v_oy/(c/2)             (as vel. increases linearly from y=0 to y=c/2)

so, dx/dt = 2v_oy/c put dt = dy/u and get

dx = (2v_o/cu)ydy    integrate, x from 0 to x, y from 0 to y, and get the eqn of trajectory as y² = cux/v_o

sudeep.kumar

Kislay... its unitary method logic...

v_x = 0 on bank, ie y=0 and
v_x = v_o at y = c/2.
For increase of c/2 in y, v_x insreases by v_o
For increase of 1 m in y, v_x will increase by v_o/(c/2) = 2v_o/c
And so for y increasing from 0 to y, v_x will increase from 0 to (2v_o/c)y....

Got it or still in doubt?? Feel free to ask again if its still not clear.

cvramana

The above solutions are correct.

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