Animated Solution for Physics - Gravitation: A body $A$ of mass $m$ is moving in a circular orbit of radius $R$ about a planet. Another body $B$ of mass $\frac{m}{2}$ collides with $A$ with a velocity which is half $\left(\frac{v}{2}\right)$ the instantaneous velocity $v$ of $A$. The collision is completely inelastic. Then, the combined body

Imagine a planet suspended in the vastness of space, and a body $A$ of mass $m$ peacefully revolving around it in a perfect circular orbit of radius $R$. To maintain this circular path, body $A$ must travel at a very specific speed, known as the orbital velocity, $v$.

Now, introduce a second body, $B$, with a mass of $\frac{m}{2}$, traveling along the exact same orbital path but at a slower pace—specifically, half the velocity of $A$, or $\frac{v}{2}$. Because $A$ is moving faster, it inevitably catches up to $B$. The problem states that the collision is completely inelastic, meaning the two bodies crash into each other and fuse into a single, combined mass.

During the brief moment of collision, the gravitational force acts radially inwards, perpendicular to the direction of motion. Because there are no external forces acting in the tangential direction, the principle of conservation of linear momentum holds true for the system of the two bodies.

Let's set up the momentum equation. The initial momentum of the system is the sum of the individual momenta of bodies $A$ and $B$:

After the inelastic collision, the bodies stick together, forming a new combined mass of $m+\frac{m}{2}=\frac{3m}{2}$. Let's call their new velocity $v_p$. The final momentum is:

Equating the initial and final momenta:

Solving for the new velocity $v_p$, we find:

A New Path

We have determined that the new velocity of the combined mass is $v_p=\frac{5}{6}v$. Notice a crucial detail here: $\frac{5}{6}v$ is strictly less than $v$.

In orbital mechanics, a circular orbit requires a very precise velocity ($v=\sqrt{\frac{GM}{R}}$). If a satellite's speed drops below this required orbital velocity, it no longer has enough kinetic energy to fight against the planet's gravitational pull at that radius. Consequently, it cannot maintain the circular path.

Instead, the combined body will begin to fall inward toward the planet, picking up speed as it loses altitude, eventually swinging around the planet and returning to the point of collision. This trajectory is an elliptical orbit, with the point of collision serving as the apogee (the farthest point from the planet).

Therefore, the combined body starts moving in an elliptical orbit around the planet, making option (b) the correct choice.