Animated Solution for Physics - Gravitation: A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth's radius $R_e$. By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion, so that it becomes $\sqrt{\frac{3}{2}}$ times larger. Due to this, the farthest distance from the centre of the earth that the satellite reaches is $R$. Value of $R$ is

A Grazing Satellite Imagine a satellite in a low circular orbit, practically skimming the Earth's surface. At this altitude, the orbital radius is essentially the Earth's radius, $R_e$. The satellite is cruising along at the standard orbital velocity, $v_0=\sqrt{\frac{GM}{R_e}}$.

Shifting Gears Suddenly, the rockets fire! The speed is instantaneously boosted by a factor of $\sqrt{\frac{3}{2}}$.

Because the speed increases instantly at this specific point, the satellite now has too much kinetic energy to remain in a circular orbit, but not enough to escape Earth's gravity entirely. The orbit stretches out, transforming into an ellipse. The point where the rockets fired becomes the closest point to Earth in this new orbit, known as the perigee. So, our perigee distance is $r_p=R_e$, and the velocity at this point is $v_p=\sqrt{\frac{3}{2}}{v}_0$.

Energy of an Ellipse To find out how far the satellite will travel, we need to look at its total mechanical energy. Energy is conserved throughout the orbit, so we can calculate it right at the perigee.

The total energy $E$ is the sum of kinetic and potential energy:

Let's substitute our boosted velocity $v_p$:

Here is a powerful secret weapon for orbital mechanics: for any elliptical orbit, the total mechanical energy is directly tied to the semi-major axis $a$ by the formula:

By equating our calculated energy to this standard formula, we can unlock the geometry of the orbit:

Canceling the common terms, we get:

The major axis ($2a$) is the total length of the ellipse, which is simply the sum of the perigee (closest distance) and the apogee (farthest distance).

We know the perigee is $R_e$, and the apogee is our unknown farthest distance $R$.

The farthest distance the satellite reaches is $3R_e$.

The Circular Orbit Trap If you look at some textbook solutions for this problem, you might see them arrive at an answer of $2R_e$. This is a classic conceptual trap!

Those flawed solutions equate the final energy of the satellite to $-\frac{GMm}{2R}$. However, $-\frac{GMm}{2R}$ is the energy of a satellite in a perfectly circular orbit of radius $R$. Our satellite is in an elliptical orbit. At its farthest point (the apogee), it is moving much slower than a satellite that is permanently circling at that distance. Always remember: an elliptical orbit has different energy characteristics than a circular orbit at the same maximum distance. Trust the math, and trust the semi-major axis!