- #1

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## Homework Statement

A satellite is put into a stationary orbit around earth, i.e. it stays above one point, then it passes through a cloud and looses 10% of it's kinetic energy. it is stabilized again in a new orbit, what's the new radius.

## Homework Equations

Attraction force: ##F=\frac{GMm}{R^2}##

Acceleration: ##a=\frac{V^2}{R}##

Relation of speed with period: ##2\pi R=V\cdot T##

## The Attempt at a Solution

Finding the initial speed by finding the initial radius:

The initial period is 24 hours, the period is ##24\cdot 3600=86,400##

$$\frac{GMm}{R^2}=\frac{V^2}{R}=\frac{4\pi^2 R^2}{T^2}$$

$$R^3=\frac{6.67E-11\cdot 6E24\cdot 86,400^2}{4\pi^2}\rightarrow R=35,904 km$$

The initial speed is:

$$S=VT\rightarrow 42,303,864=V\cdot 86,400\rightarrow V=3076$$

The initial kinetic energy:

$$\frac{1}{2}m\cdot 3076^2=4,257,799$$

The final velocity after deducing 10%:

$$4,257,799\cdot 0.9 \cdot m=\frac{1}{2}m\cdot V^2\rightarrow V^2=8,515,598$$

$$R=\frac{GM}{V^2}=\frac{6.67E-11\cdot 6E24}{8,515,598}=46,996 km$$

It should be R=40,500 km