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Question: A bullet of mass \(0.01kg\) is fired from a gun weighing \(5kg\) if the initial speed of the bullet ...

A bullet of mass 0.01kg0.01kg is fired from a gun weighing 5kg5kg if the initial speed of the bullet is 250mps250mps. Calculate speed with which the gun recoils.
a)−0.50ms−1 b)−0.25ms−1 c)+0.05ms−1 d)+0.25ms−1 \begin{aligned} & a)-0.50m{{s}^{-1}} \\\ & b)-0.25m{{s}^{-1}} \\\ & c)+0.05m{{s}^{-1}} \\\ & d)+0.25m{{s}^{-1}} \\\ \end{aligned}

Explanation

Solution

Conservation of angular momentum plays an important role in this case. For every action, there’s an equal and opposite reaction. Therefore, if we can find the momentum of the bullet and equalize it with the momentum of the gun, we can find the speed of the gun recoil.

Formulas used:
mgvg+mbvb=0{{m}_{g}}{{v}_{g}}+{{m}_{b}}{{v}_{b}}=0

Complete step-by-step solution:
Let us assume the mass of the gun, the velocity of the gun as mg,vg{{m}_{g}},{{v}_{g}} respectively, and mass of the bullet and velocity of the bullet as mb,vb{{m}_{b}},{{v}_{b}} respectively.
Now, the law of angular momentum is conserved,
mgvg+mbvb=0 5×vg+(0.01)×250=0 vg=−0.5ms−1 \begin{aligned} & {{m}_{g}}{{v}_{g}}+{{m}_{b}}{{v}_{b}}=0 \\\ & 5\times {{v}_{g}}+(0.01)\times 250=0 \\\ & {{v}_{g}}=-0.5m{{s}^{-1}} \\\ \end{aligned}
Therefore, the correct option is option a.

Additional information:
The law of conservation of momentum states that the momentum of an isolated system remains constant. The momentum of the whole system is neither created nor destroyed, but only can be transferred from one form to another form. This law can be proven by Noether’s theorem. This law was first stated by Rene Descartes. This law of conservation of momentum holds true for almost all the particles and interactions including collisions and separations caused due to explosions. It can also be generalized when Newton's laws don’t hold true, as the theory of relativity and electrodynamics.
The law of conservation of momentum also holds good for quantum mechanics. In those phenomena when particles are manifested, their momentum is similar to the momentum of particles in classical mechanics. In the quantum world, the law of conservation of momentum is a consequence of symmetry with respect to shift in space.

Note: The speed of the gun when it recoils is negative because the gun moves in the negative x-axis if we assume the direction of the bullet as the positive x-axis. If so, there is any friction, the momentum is not conserved. The conservation of momentum is also called Newton's third law of motion.