Animated Solution for Physics - System of Particles: A bomb of mass $16\text{ kg}$ at rest explodes into two pieces of masses $4\text{ kg}$ and $12\text{ kg}$. The velocity of the $12\text{ kg}$ mass is $4{\text{ ms}}^{-1}$. The kinetic energy of the other mass is

Imagine a heavy $16\text{ kg}$ bomb sitting perfectly still on the ground. Because it is completely at rest, its initial velocity is zero. Consequently, its initial momentum is also zero.

Suddenly, an internal chemical reaction triggers an explosion! The bomb violently splits into two distinct pieces. One piece, weighing $12\text{ kg}$, flies off to the right at a speed of $4{\text{ ms}}^{-1}$. The other piece, weighing $4\text{ kg}$, shoots off in the opposite direction with an unknown velocity, which we will call $v$.

In physics, an explosion is a classic example of a system driven entirely by internal forces. Because there are no external forces (like friction or an outside push) acting on the bomb during the split, the total linear momentum of the system must remain perfectly conserved.

This means the momentum before the explosion must exactly equal the momentum after the explosion:

Since the bomb was initially at rest, the initial momentum is zero. Therefore, the vector sum of the momenta of the two flying pieces must also add up to zero. This is exactly why the two pieces must fly in opposite directions!

Let's set up our mathematical equation. We will define the rightward direction as positive and the leftward direction as negative.

Substituting our known values into the equation:

Now, we simply move the $-4v$ to the other side of the equation:

We have successfully found that the $4\text{ kg}$ mass is flying off to the left at a rapid $12\text{ m/s}$.

The question ultimately asks for the kinetic energy of this $4\text{ kg}$ mass. The formula for kinetic energy is one of the most famous in classical mechanics:

Let's carefully substitute our mass and the velocity we just calculated:

First, we square the velocity: $12^2=144$. Next, we take half of the mass: $\frac{1}{2}\times 4=2$. Finally, we multiply them together:

And there we have it! The kinetic energy of the $4\text{ kg}$ mass is $288\text{ J}$.