let’s discuss the concept of reaction due
to ejection of a liquid. if we just talk about
a situation in which say we are having a cylindrical
container, and say it is placed on ground.
and in this cylindrical container, say a liquid
is filled. and, at any point in the side walls,
of this cylindrical container if we make a
hole, say we are also given that cross sectional
area of hole is a. then fluid will start ejecting
out with the efflux velocity v. now we can
state, initially the whole system was at rest,
and if the fluid is ejected out with some
speed. that means we can say, the ejecting
out fluid will be having some momentum. so,
according to newton’s 3rd law, or according
to the concept of conservation of momentum
we can say, the container, along with remaining
fluid must gain some momentum in backward
direction so that the overall momentum of
system remains, zero. now in this situation,
if we find out the flow rate, of liquid, ejecting
out is, this flow rate can be written as,
d v by d t, is equal to, the product of, cross
sectional area and the speed with which it
is coming out. and if we find out the mass
rate, of ejecting liquid is, this mass rate
d m by d t can be simply written as, ro a-v,
if ro is the density, of the liquid, then
ro multiplied by the volume flow rate is the
mass rate of ejection. now in this situation
if we find out, the momentum, ejecting, per
second, in fluid is, the momentum ejecting
per second in fluid can be written as the
force, because the rate at which momentum
is ejecting out, this container will experience
a reaction force in backward direction. so
this force can be written as, ro a-v multiplied
by the speed because the mass rate of ejection,
multiplied by velocity is the rate at which
momentum is ejecting out, is ro a-v square.
so we can simply state, the force of reaction,
on container, due to ejection of liquid is,
this reaction force can be written as, ro
a-v square. always keep this result on tips,
as in many applications we will have to use
it.
