Animated Solution for Physics - System of Particles: A body of mass $2\text{ kg}$ moving with a speed of $4\text{ m/s}$ makes an elastic collision with another body at rest and continues to move in the original direction but with one-fourth of its initial speed. The speed of the two body centre of mass is $\frac{x}{10}\text{ m/s}$. Then, the value of $x$ is ......... .

The problem of collisions is one of the most fascinating areas in classical mechanics. It’s where the abstract laws of conservation suddenly dictate the chaotic crashing of objects. In this problem, we are tasked with finding the speed of the center of mass of a two-body system after an elastic collision. Let's break down the physics step-by-step.

The Collision

Imagine a body of mass $m_1=2\text{ kg}$ cruising along at a speed of $u_1=4\text{ m/s}$. It’s on a direct collision course with a second body of unknown mass $m_2$ that is currently at rest ($u_2=0\text{ m/s}$).

The problem tells us that after the collision, the first body continues in its original direction but at one-fourth of its initial speed. This is a massive clue! We can immediately calculate its final velocity:

In any collision where no external horizontal forces are acting, the total linear momentum of the system is strictly conserved. This means the momentum before the crash must perfectly equal the momentum after the crash.

Let's substitute the values we know into this master equation:

We now have a relationship between the mass and final velocity of the second body, but we need another equation to solve for them individually.

The problem explicitly states that the collision is elastic. In physics, an elastic collision is a special type of impact where kinetic energy is perfectly conserved. Mathematically, this is represented by the coefficient of restitution ($e$) being exactly equal to $1$.

The coefficient of restitution relates the relative velocities before and after the collision:

This tells us that the relative speed of separation equals the relative speed of approach. Let's plug in our velocities:

Now that we know the second body shoots off at $5\text{ m/s}$, we can revisit our momentum equation to find its mass:

The System's Heartbeat

We have all the pieces of the puzzle. The final question asks for the speed of the center of mass of the two-body system. The center of mass is a unique point that moves as if all the system's mass were concentrated there and all external forces were applied to it.

The velocity of the center of mass ($v_{CM}$) is given by the total momentum divided by the total mass:

Let's substitute our known values:

The problem states that the speed of the center of mass is $\frac{x}{10}\text{ m/s}$. We just need to equate our result to this expression to find $x$:

And there we have it! By carefully applying the laws of conservation of momentum and the definition of an elastic collision, we've successfully navigated the problem.