64. Physics | Kinematics | Two Perpendicular Trips in a River | by Ashish Arora (GA)

this illustration, is analyzing 2 perpendicular
trips in a river. we are given that 2 identical boats ay and b start from a point in river
with speed v relative to water in mutually perpendicular paths. boat ay along the river
current and boat b perpendicular to river current. here if we are given that if they
move equal distances from initial point and return back we are required to find the ratio
of time taken by the boats ay and b is round trips. and the velocity of river current is
given as point 8 v. so here if we analyze the situation say if this is a river, and
say a boat, ay is moving along the river current. and the speed of ay here we can directly write
if river current speed is u. which is given to us as point 8 v, where v is, the boat speed
with respect to river. so ay will be moving with the speed, v plus u. and after travelling
some distance if it return back, its speed with respect to ground will be, v minus u,
and it will come back. and we are given if boat b is moving perpendicular to river current
then boat b is moving in this direction it, goes the same distance, and then it return
back, and coming back to the same point. so in this situation for going, it has to travel
with the speed v in some direction so that when u added the resultant path would be normal,
similarly when it is returning its velocity would be. somewhere here or you can say in
this direction, so that, when u will be added, the resulting velocity will be. in the direction
normal to river current. so here we can write, for, distance l. travelled by. boats. as we
assume, ay will travel a distance l and b will also go and come back for a distance
l. we can write time taken. by ay is. here ay is travelling with speed v plus u and then
return as with v minus u so time for ay we can write as l by v plus u plus, l by v minus
u. on simplifying we are getting it 2 v l by v square minus, u square. similarly for
b as it is going with, speed v and, river current speed u vectorially added to it the
resultant will be taken as root of v square minus u square and the same will be in, return
path, so we can write the time taken. by b is. here t b can be written as this l by root
of v square minus u square plus. l by root of v square minus u square that is 2 l by,
root of v square minus u square. so if we calculate the ratio of the 2 times t ay by
t b. on substituting the values you can see it is 2 v l upon v square minus u square.
divided by 2 l upon root of v square minus u square here 2 l gets cancelled out. and
on simplifying we are getting it as v by. root of v square minus u square if we substitute
the value of river current speed. so this will be v divided by root of v square minus.
u is zero point 8 v so if we square it this will be zero point 6 4 v square. so on simplifying,
we can see here v also gets cancelled out and this 1 by root of zero point. 3 6, so
that will be 1 by zero point 6 and, we are getting, this as, 5 by 3. so, this will be
the final result of the problem the ratio of the time taken by boats ay and b in travelling
equal distances as described in the problem.

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