Animated Solution for Physics - System of Particles: A bullet of $4\text{ g}$ mass is fired from a gun of mass $4\text{ kg}$. If the bullet moves with the muzzle speed of $50{\text{ ms}}^{-1}$, the impulse imparted to the gun and velocity of recoil of gun are

Imagine you are standing on a frictionless surface holding a heavy rifle. When you pull the trigger, the bullet shoots forward with immense speed. But you don't just stand still—you feel a sharp, powerful kick backward. This backward kick is what physicists call recoil.

In this problem, we are going to explore the beautiful mechanics behind this phenomenon using two fundamental concepts: the Conservation of Linear Momentum and Impulse.

We have a gun of mass $m_G=4\text{ kg}$ and a bullet of mass $m_B=4\text{ g}$. Before we do any math, we must ensure our units are consistent. Let's convert the bullet's mass to kilograms:

The bullet is fired with a muzzle speed of $v_B=50{\text{ ms}}^{-1}$. We need to find two things: the recoil velocity of the gun ($v_G$) and the impulse imparted to it.

Conservation of Momentum

Why does the gun move backward? Because of the Law of Conservation of Linear Momentum. Since there are no external horizontal forces acting on the gun-bullet system (like friction or an external push), the total momentum of the system must remain constant.

Initially, before the trigger is pulled, both the gun and the bullet are at rest. Therefore, the initial total momentum is zero.

After firing, the bullet gains forward momentum, and the gun must gain an equal amount of backward momentum so that the total sum remains zero.

Equating the initial and final momentum:

Let's substitute our known values into the master equation.

Now, we solve for $v_G$:

The negative sign is crucial here! It mathematically proves what we intuitively know: the gun moves in the opposite direction to the bullet.

Next, we need to find the impulse imparted to the gun. What exactly is impulse? Impulse ($I$) is defined as the change in momentum of an object.

For the gun, the initial momentum was zero. The final momentum is its mass times its recoil velocity.

The magnitude of this impulse is $0.2{\text{ kg-ms}}^{-1}$.

We have successfully decoded the physics of the gunshot. The impulse imparted to the gun is $0.2{\text{ kg-ms}}^{-1}$, and the magnitude of the recoil velocity is $0.05{\text{ ms}}^{-1}$. This perfectly matches option (b).

Physics isn't just about plugging numbers into formulas; it's about understanding the invisible rules that govern every action and reaction in our universe!