Answer:
The orbital speed of Dactyl is [tex]5.55m/s[/tex]
Explanation:
The orbital speed can be determined by the combination of the universal law of gravity and Newton's second law:
[tex]F = G\frac{M \cdot m}{r^{2}}[/tex] Â (1)
Where G is gravitational constant, M is the mass of the asteroid, m is the mass of the moon and r is the distance between them
In the other hand, Newton's second law can be defined as:
[tex]F = ma[/tex] Â (2)
Where m is the mass and a is the acceleration
Then, equation 2 can be replaced in equation 1
[tex]m\cdot a  = G\frac{M \cdot m}{r^{2}}[/tex]  (2)
However, a will be the centripetal acceleration since the moon Dactyl describe a circular motion around the asteroid
[tex]a = \frac{v^{2}}{r}[/tex] Â (3)
[tex]m\frac{v^{2}}{r} = G\frac{M \cdot m}{r^{2}}[/tex] (4)
Therefore, v can be isolated from equation 4:
[tex]m \cdot v^{2} = G \frac{M \cdot m}{r^{2}}r[/tex]
[tex]m \cdot v^{2} = G \frac{M \cdot m}{r}[/tex]
[tex]v^{2} = G \frac{M \cdot m}{rm}[/tex]
[tex]v^{2} = G \frac{M}{r}[/tex]
[tex]v = \sqrt{\frac{G M}{r}}[/tex] (5)
Finally, the orbital speed can be found from equation 5:
Notice, that it is necessary to express r in units of meters.
[tex]r = 95km \cdot \frac{1000m}{1km}[/tex] ⇒ [tex]95000m[/tex]
[tex]v = \sqrt{\frac{(6.672x10^{-11}N.m^{2}/kg^{2})(4.4x10^{16}kg)}{95000m}}[/tex]
[tex]v = 5.55m/s[/tex]
Hence, the orbital speed of Dactyl is [tex]5.55m/s[/tex]
