21. Physics | Newton’s Laws of Motion | A Motor Boat in a Lake | by Ashish Arora (GA)


this illustration is about a motor boat in
a lake. here we are given that a motor boat of mass m, moves in a lake with velocity v
not. at t equal to zero the engine of boat is switched off. and the water resistance
is proportional to speed of boat and, it varies as f r is minus r v. where r is a constant.
and we are required to find the time for which boat will move before coming to rest. and
also we are required to find the average velocity of boat during the interval its velocity changes
from v not to v not by 2. so here. first we can directly write, that the acceleration
of boat. can be written as ay is equal to minus r by m, multiplied by v. because in
this situation we are given that resistance force is minus r v. so in this situation we
can write ay as d v by d t so. d v by d t is minus r by m, times v. which can be written
as d v by v is equal to minus r by, m d t. so if we integrate this. term, at t equal
to zero the speed was v not at, time t. the speed changes to v. so lefthand side becomes
ellen of v. i am substituting limits this will be ellen of v by v not. is equal to minus
r t by, m. on simplifying this will give us the speed as v not e to power minus, r t by
m. so here we can see the value of v is equal to zero. when, here t is tending to infinity,
when this t tends to infinity then only the speed. become zero. and here we are required
to find the time for which boat will move before coming to rest. so v equal to zero
which signifies rest. so boat will come to rest only when, t tending to infinity after
very very long time this will be the result of the first part of the problem. and here
we are also required to find, the average velocity of boat during time interval its
velocity changes from v not to v not by 2. so if we continue, here. we can write, at
t equal to zero the value of speed was v not. and we can write at t is equal to t 1. if
speed changes to v not by 2. then using this expression, we can find out the time t 1.
so if we write this as equation 1 then we can write from 1. we can substitute v as v
not by 2 so this v not by 2 is equal to. v not e to power minus r t by, m. and the value
of t we can put as t 1. so in this situation this v not gets cancelled out and on lefthand
side it can be written as, ellen 2 is equal to r t 1 by m this implies the value of time.
after which, the speed reduces to v not by 2. is, m by, r. ellen 2. so now, we can directly
calculate the average velocity by the average function which is given as 1 by t 1 integration
from zero to t 1. this v as a function of time d t. on substituting the value this is
1 by t 1 integration zero to t 1. and v as a function of time as v not e to power minus
r t by, m. d t. so if we integrate this term, the average velocity here can be given as,
now on integrating this term it is v not by t 1 and if we integrate e to power minus r
t by m. this will be v not m by, r t 1. and its integration remain same this is minus
e to power, minus r t by, m. and limits from zero to t 1. so if we substitute the value
of t 1 it is m by r ellen 2. so the lefthand side here it becomes, v not, by ellen 2. and
this term if we substitute t 1 this will become. minus e to power minus r by m t 1 which can
be written as minus ellen 2. and if we substitute zero this will become plus 1 so we can write
it 1 minus, e to power minus ellen 2. and, this can be further written as e to power
minus ellen 2 it can be written as 1 by 2, so the result is v not by 2 ellen 2, that
will be the final result of the problem.

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