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**Hint:**In the above given question we have to find the recoil velocity of the gun and we are given the mass of the bullet and the speed of the bullet so here we will use the formula of principle of conservation of momentum to find the recoil velocity of the gun.

**Complete step-by-step solution:**

The given here question says that the mass of the bullet is 0.1 kg and the speed of the fired bullet is 100 \[m{{s}^{-1}}\]. We have to find the recoil velocity of the gun. For solving the given question, we will use the principle of conservation of momentum

So, let us first discuss what principle of conservation of momentum says:

The principle of conservation of momentum says that if a collision occurs between object 1 and object 2 in any isolated system, the total momentum of the two objects before the collision is always equal to the total momentum of the two objects after the collision. That is, the momentum lost by object 1 is equal to the momentum gained by object 2. Therefore we can say that momentum can neither be created nor be destroyed; it can only be transformed from one object to another.

Mathematically principle of conservation of momentum is,

Initial momentum = final momentum

Or, the momentum of gun = momentum of bullet

\[{{m}_{1}}{{v}_{1}}={{m}_{2}}{{v}_{2}}\]

Where \[{{m}_{1}}\]is the mass of the gun

\[{{v}_{1}}\]is the recoil velocity of the gun

\[{{m}_{2}}\]is the mass of the bullet

\[{{v}_{2}}\]is the recoil velocity of the bullet

Substituting the given values in the equation:

\[{{m}_{1}}\]is the mass of the gun here it is 50 kg

\[{{v}_{1}}\]is the recoil velocity of the gun here we have to find it

\[{{m}_{2}}\]is the mass of the bullet here it is 0.1 kg

\[{{v}_{2}}\]is the velocity of the bullet here it is 100 \[m{{s}^{-1}}\]

\[50\times {{v}_{1}}=0.1\times 100\]

\[{{v}_{1}}\]= \[\dfrac{0.1\times 100}{50}\]

=0.2 \[m{{s}^{-1}}\]

So, the recoiling velocity of gun is0.2 \[m{{s}^{-1}}\]

**Note:**In questions like these where we have different objects of different sizes always keep in mind the units of the quantities given to us in the question as the mass of the bullet being very small can be in grams rather than being in kilograms that may result in wrong answers for us.

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