Question

A satellite is in sufficiently low obit so that it encounter air drag and if orbit changes from r to r - Dr. 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. $\frac{\Delta r}{2}\sqrt{\frac{GM}{r^{2}}},\frac{GMm}{r}$
• C. $\frac{\Delta r}{2}\sqrt{\frac{GM}{r^{3}}},\frac{GMm\Delta r}{2r^{2}}$
• D. None

Answer 1

Answer: (A)

$\frac{\Delta r}{2}\sqrt{\frac{GM}{r^{3}}},\frac{GMm\Delta r}{r^{2}}$

Answer 2

v₀$\sqrt{\frac{GM}{r}}$ ; v₀' = $\sqrt{\frac{GM}{r - \Delta r}}$

= $\sqrt{\frac{GM}{r}}\left( 1 - \frac{\Delta r}{r} \right)^{- 1/2} = \sqrt{\frac{GM}{r}}\left( 1 + \frac{\Delta r}{2r} \right)$∆v = v₀' – v₀ = $\sqrt{\frac{GM}{r}}$

∆U = -2∆KE = - $\left( \frac{1}{2}m \right)$ [v₀'² - v₀²]

= m$\left\lbrack \frac{GM}{r - \Delta r} - \frac{GM}{r} \right\rbrack = \frac{GMm}{r}\left\lbrack \frac{\Delta r}{r} \right\rbrack = \frac{GMm\Delta r}{r^{2}}$