The centrifugal force
Let us consider a point mass  for reasons of simplicity  on a string
swinging around in a circle (e.g. in vacuum, with constant speed). This
mass experiences the centripetal force F_{p} which
is directed inward, towards the center of the circle.
However the hand, holding the string "feels" an outward directed force
 according to Newton´s third law (actio=reactio). This "reactive" forceF_{R}
has the same magnitude as the centripetal force  but opposite direction
 and applies at the hand holding the string. This reactive force is not
the centrifugal force, because the centrifugal force applies at the center
of mass of the body.
The necessity to consider a centrifugal force can best be understood
from the following:
Assume a mass on a spring that swings around in a circle with constant
velocity. This spring will show some tension.
From the position of an observer in a reference system which rotates
with the mass one finds:

the mass does not move,

there is a tension in the spring
This tension could only be explained by the existence of an outward
directed force  the centrifugal force.
The necessity to consider the centrifugal force simply arises from the
fact, that the reference system fixed to the earth, within which we consider
bullet motion is rotating.
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