Question

A satellite is in sufficiently low orbit so that it encounter air drag and if orbit changes from r to $(r-\Delta r)$. Find the change in orbital velocity and change in PE.

• A. $\frac{\Delta r}{2}\sqrt{\frac{GM}{r^3}},\frac{GMm\Delta r}{r^2}$
• B. $\frac{\Delta r}{2}\sqrt{\frac{GM}{r^2}},\frac{GMm}{2r^2}$
• C. $\frac{\Delta r}{2}\sqrt{\frac{GM}{r^3}},\frac{GMm\Delta r}{2r^2}$
• D. none

Answer 1

The correct option is A $\frac{\Delta r}{2}\sqrt{\frac{GM}{r^3}},\frac{GMm\Delta r}{r^2}$

$v_0=\sqrt{\frac{GM}{r}};v_0^{\prime }=\sqrt{\frac{GM}{r-\Delta r}}$
$=\sqrt{\frac{GM}{r}}{(1-\frac{\Delta r}{r})}^{-1/2}=\sqrt{\frac{GM}{r}}(1+\frac{\Delta r}{2r})$
$\Delta v=v_0^{\prime }-v_0=\frac{\Delta r}{2r}\sqrt{\frac{GM}{r}}$.
$\Delta U=-2\Delta KE=-2\left(\frac{1}{2}m\right)[v_0^{\prime 2}-v_0^2]$
$=m[\frac{GM}{r-\Delta r}-\frac{GM}{r}]=\frac{GMm}{r}\left[\frac{\Delta r}{r}\right]$
$=\frac{GMm\Delta r}{r^2}$