Animated Solution for Physics - System of Particles: Two bodies of the same mass are moving with the same speed, but in different directions in a plane. They have a completely inelastic collision and move together thereafter with a final speed which is half of their initial speed. The angle between the initial velocities of the two bodies (in degree) is .............

A Symmetrical Crash

Imagine two identical bodies, each possessing a mass $m$, hurtling towards each other with the exact same speed $v$. They aren't moving head-on, but rather at an angle to one another. When they collide, it's a completely inelastic collision—meaning they stick together and move as a single, combined mass of $2m$.

Because the masses and speeds are identical, the collision is perfectly symmetrical. This symmetry is our greatest weapon. It tells us that the final combined mass will move exactly along the angle bisector of their initial paths. Let's define this line of motion as our x-axis. Consequently, each initial velocity vector makes an equal angle, let's call it $\alpha$, with this x-axis.

In any collision, whether objects bounce off each other perfectly or mash together into a single lump, one fundamental law of physics always holds true: the conservation of linear momentum. As long as no external forces are acting on the system, the total momentum before the crash must equal the total momentum after the crash.

Mathematically, we write this as:

Components are Key

To apply this law, we need to break the initial velocities into their x and y components.

For the first body, the momentum is $m(v\cos \alpha \hat{i}-v\sin \alpha \hat{j})$. For the second body, the momentum is $m(v\cos \alpha \hat{i}+v\sin \alpha \hat{j})$.

When we add these together to find the total initial momentum ${\vec{p}}_i$, something beautiful happens. The y-components are equal and opposite, so they perfectly cancel each other out! We are left with only the x-components:

Now, let's look at the final momentum ${\vec{p}}_f$. The bodies have stuck together, creating a mass of $2m$. The problem tells us their final speed is half of their initial speed, which is $v/2$. Since they move along the x-axis, the final momentum is:

Solving for the Angle

Now we simply equate the initial and final momenta:

The $m$ and $v$ terms gracefully cancel out from both sides, leaving us with a very simple trigonometric equation:

We know from basic trigonometry that the angle whose cosine is $1/2$ is $60^{\circ }$. Therefore, $\alpha =60^{\circ }$.

But we aren't quite done! The question asks for the total angle between the initial velocities. Since each velocity vector makes an angle $\alpha$ with the x-axis, the total angle $\theta$ between them is $2\alpha$.

And there we have it! The two bodies must have collided at an angle of $120^{\circ }$ to achieve a final speed that is exactly half of their initial speed.