Animated Solution for Physics - Gravitation: A satellite is in an elliptical orbit around a planet $P$. It is observed that the velocity of the satellite when it is farthest from the planet is 6 times less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is

Imagine a satellite gracefully sweeping through the vacuum of space in an elliptical orbit around a massive planet. According to Kepler's First Law, the planet doesn't sit at the center of this ellipse, but rather at one of its focal points. This geometric reality means the satellite's distance from the planet is constantly changing.

There are two very special points in this orbit: 1. The Perigee (Closest Point): Here, the satellite is at its minimum distance, $r_{\text{min}}$. Because it has fallen deep into the planet's gravitational well, it whips around at its maximum speed, $v_{\text{max}}$. 2. The Apogee (Farthest Point): Here, the satellite has climbed out of the gravity well to its maximum distance, $r_{\text{max}}$. Exhausted from the climb, it moves at its minimum speed, $v_{\text{min}}$.

Conservation of Angular Momentum

Why does the satellite speed up and slow down so predictably? The secret lies in the nature of gravity. Gravity is a central force—it always pulls the satellite directly toward the center of the planet.

Because the force vector and the position vector are parallel, the torque ($\tau =\vec{r}\times \vec{F}$) acting on the satellite is exactly zero. When there is no net torque, a system's angular momentum ($L$) must remain perfectly conserved.

The angular momentum of a particle is given by $L=mvr\sin \theta$. At the exact points of perigee and apogee, the velocity vector is perfectly perpendicular to the position vector ($\theta =90^{\circ }$), making $\sin (90^{\circ })=1$.

Thus, we can write the conservation equation simply as:

Notice how the mass of the satellite, $m$, appears on both sides? We can cancel it out entirely. This tells us a beautiful truth: the orbital geometry depends only on the velocities and distances, not on how heavy the satellite is!

Rearranging the equation to group our distances and velocities, we get:

The problem hands us a crucial piece of intel: the velocity at the farthest point is 6 times less than at the closest point. Mathematically, this translates to:

Or, written as a ratio:

By substituting this velocity ratio directly into our angular momentum equation, the answer reveals itself instantly:

The ratio of the closest distance to the farthest distance is exactly $1:6$.