Animated Solution for Physics - System of Particles: Two particles of equal mass $m$ have respective initial velocities $u\hat{i}$ and $u\left(\frac{\hat{i}+\hat{j}}{2}\right)$. They collide completely inelastically. The energy lost in the process is

In the fascinating world of mechanics, collisions are a perfect playground to test our understanding of conservation laws. When two objects collide, the outcome depends heavily on the nature of the collision. In this problem, we are dealing with a completely inelastic collision, a scenario where objects stick together upon impact, moving as a single entity afterwards.

Let's break down the physics and mathematics behind this event step-by-step.

The Collision Course

Imagine two particles, each possessing a mass $m$, moving in a two-dimensional plane. The first particle is moving purely along the x-axis with a velocity vector:

The second particle is moving at an angle, with its velocity vector given by:

These two particles are on a collision course. Because the collision is completely inelastic, they will merge into a single combined mass of $2m$ moving with some final velocity $\vec{v}$.

Conservation of Momentum

In any collision where external forces are absent, the total linear momentum of the system is strictly conserved. This is our master key to unlocking the final velocity. We equate the initial momentum of the system to its final momentum:

Substituting the momenta of our particles, we get:

Notice how the mass $m$ is common to all terms. We can elegantly cancel it out, simplifying our equation to:

Combining the $\hat{i}$ components, we find:

Dividing the entire equation by 2 yields the final velocity vector of the combined mass:

To find the energy lost during the collision, we must calculate the kinetic energy before and after the impact. Remember, kinetic energy is a scalar quantity defined as $K=\frac{1}{2}m{v}^2$. For a velocity vector, the square of the speed $v^2$ is simply the sum of the squares of its orthogonal components ($v^2=v_x^2+v_y^2$).

Let's calculate the initial kinetic energy ($K_i$), which is the sum of the kinetic energies of the two individual particles:

Now, let's calculate the final kinetic energy ($K_f$) of the combined mass $2m$ moving with velocity $\vec{v}$:

Energy Lost

In a completely inelastic collision, maximum kinetic energy is dissipated (usually as heat, sound, or deformation energy). The energy lost ($\Delta K$) is the difference between the initial and final kinetic energies:

To subtract these fractions, we find a common denominator of 8:

This elegant result shows exactly how much of the system's initial kinetic energy was sacrificed to bind the two particles together!