Question

Calculate the speed of a satellite moving in a stable circular orbit about the Earth at a height of 4800 km.

Short answer

The speed of a satellite is $5.97\times {10}^3\mathrm{m}/\mathrm{s}$.

Step-by-step solution

Step 1. Given Data

The height is $h=4800\mathrm{km}$.

Step 2. Understanding the speed of the satellite

In this problem, utilize the relation of the orbital speed in terms of mass of the certain body and its distance from the satellite to the centre for calculating the speed of a satellite.

Step 3. Estimating the speed of the satellite

The relation of orbital speedis given by,

$v=\sqrt{\frac{G{m}_{\mathrm{E}}}{r_{\mathrm{E}}+h}}$

Here, G is the gravitational constant, $r_{\mathrm{E}}$is the radius of the Earth and $m_{\mathrm{E}}$is the mass of the Earth.

On plugging the values in the above relation.

$\begin{array}{l}v=\sqrt{\left[\frac{\left(6.67\times {10}^{-11}\mathrm{N}·{\mathrm{m}}^2/{\mathrm{kg}}^2\right)\left(5.98\times {10}^{24}\mathrm{kg}\right)}{\left(6.38\times {10}^6\mathrm{m}\right)+\left(4800\mathrm{km}\times \frac{1000\mathrm{m}}{1\mathrm{km}}\right)}\right]}\\ v=5.97\times {10}^3\mathrm{m}/\mathrm{s}\end{array}$

Thus, $v=5.97\times {10}^3\mathrm{m}/\mathrm{s}$is the required speed.