- #1
Karol
- 1,380
- 22
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