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**Hint:**The bullet will eventually come to a state of rest. In this state of rest there will be no motion of the bullet. This means that the velocity of the bullet when it is ion rest state will be zero. thus the final velocity of the bullet will be zero. The displacement is given i.e. the thickness of the tree trunk. Use the equation of motion to find the retardation.

**Complete step by step solution:**

Motion equations are equations that characterise the actions of a physical system as a function of time in terms of movement. More precisely, movement equations characterise physical system behaviour in terms of dynamic variables as a series of mathematical functions. These variables typically contain spatial coordinates, time and momentum. The general alternative are widespread coordinates that are appropriate for any comfortable physical device variables. The functionalities in classical mechanics are described in a Euclidean space but are replaced in relativity by curved spaces. The equations are the solutions for the difference equation which describes the motion of the dynamics if the dynamics of a system are understood.An object's travel can take several directions. We will concentrate here on movement in a straight line (one dimension). Therefore, we can only use positive and negative dimensions of displacement, speed and acceleration where negative values are opposite to positive numbers.

Initial Velocity of the bullet is $u = 16m{s^ {- 1}} $

Final velocity of the bullet is zero, because the bullet penetrates a tree trunk and comes to rest.

Distance, $s = 0.4m$

Hence, using the third equation of motion,

${v^2} - {u^2} = 2as$

$ \Rightarrow 0 - {(16) ^2} = 2a \times 0.4$

$a = \dfrac{{16 \times 16}} {{2 \times 0.4}} $

Retardation is $320m{s^ {- 2}} $

Using the first equation of motion,

$v = u + at$

$ \Rightarrow 0 = 16 - 320t$

$t = \dfrac{{16}}{{320}}$

$t = 0.05s$

**The time taken for retardation is $0.05s$.**

**Note:**According to mechanics when a bullet is fired from a gun, the bullet is given some initial velocity with which it undergoes a uniform motion with the velocity being constant. Thus there is no acceleration or retardation upto a certain limit. This limit is the range of the gun upto which the fired bullet undergoes uniform motion. However, if during this motion some external force acts upon the bullet then there comes an acceleration. Mostly this acceleration is in the opposite direction to the velocity and so it is called retardation. So, the bullet gains a retardation after striking or colliding with the trunk of the tree.

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