Animated Solution for Physics - Laws of Motion: A machine gun fires a bullet of mass $40\text{ g}$ with a velocity $1200{\text{ ms}}^{-1}$. The man holding it, can exert a maximum force of $144\text{ N}$ on the gun. How many bullets can he fire per second at the most?

Understanding Recoil

Imagine you are holding a powerful machine gun. You pull the trigger, and a hail of bullets flies out of the barrel at supersonic speeds. But as the bullets rush forward, you feel a violent, backward shove against your shoulder. This is the recoil, and it is one of the most visceral demonstrations of Newton's Laws of Motion in action.

To understand why this happens, we have to look at the fundamental rules that govern our universe, specifically Newton's Third Law of Motion and the concept of momentum.

Newton's Third Law states that for every action, there is an equal and opposite reaction. When the gunpowder explodes inside the gun, it pushes the bullet forward. Simultaneously, the bullet pushes the gun backward with the exact same force.

We can quantify this using momentum. Momentum ($p$) is the product of an object's mass and its velocity:

In our problem, a single bullet has a mass of $40\text{ g}$. In physics, we must always work in standard SI units, so we convert this to kilograms:

The bullet leaves the barrel with a staggering velocity of $1200\text{ m/s}$. Therefore, the momentum carried away by a single bullet is:

Force as a Rate of Change

Now, how does this momentum translate into the force you feel? Newton's Second Law of Motion tells us that force is the rate of change of momentum:

If the machine gun fires $n$ bullets every second, the total momentum leaving the gun per second is $n\times p$. Because this momentum is generated every single second, it perfectly equals the continuous force exerted on the gun:

The problem states that the man holding the gun can exert a maximum forward force of $144\text{ N}$ to keep the gun steady. If the recoil force exceeds this limit, the gun will overpower him and slip from his grasp.

To find the maximum number of bullets he can fire per second without losing control, we set the recoil force equal to his maximum holding force:

Solving for $n$, we get:

The math reveals a strict physical limit. The man can fire a maximum of 3 bullets per second.

If he were to fire 4 bullets per second, the recoil force would jump to $192\text{ N}$, completely overwhelming his $144\text{ N}$ grip. This elegant interplay between mass, velocity, and force is what makes classical mechanics so incredibly powerful and predictive.