Question

Asteroid Ida was photographed by the Galileo spacecraft in 1993, and the photograph revealed that the asteroid has a small moon, which has been named Dactyl. Assume the mass of Ida is 4.4 x 1016 kg, the mass of Dactyl is 2.6 x 1012 kg, and the distance between the center of Dactyl and Ida is 95 km. G = 6.672x10-11 N-m2/kg2 Part Description Answer Save Status A. Assuming a circular orbit, what would be the orbital speed of Dactyl? (include units with

Answer 1

Answer:

The orbital speed of Dactyl is $5.55m/s$

Explanation:

The orbital speed can be determined by the combination of the universal law of gravity and Newton's second law:

$F = G\frac{M \cdot m}{r^{2}}$  (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:

$F = ma$  (2)

Where m is the mass and a is the acceleration

Then, equation 2 can be replaced in equation 1

$m\cdot a = G\frac{M \cdot m}{r^{2}}$  (2)

However, a will be the centripetal acceleration since the moon Dactyl describe a circular motion around the asteroid

$a = \frac{v^{2}}{r}$  (3)

$m\frac{v^{2}}{r} = G\frac{M \cdot m}{r^{2}}$ (4)

Therefore, v can be isolated from equation 4:

$m \cdot v^{2} = G \frac{M \cdot m}{r^{2}}r$

$m \cdot v^{2} = G \frac{M \cdot m}{r}$

$v^{2} = G \frac{M \cdot m}{rm}$

$v^{2} = G \frac{M}{r}$

$v = \sqrt{\frac{G M}{r}}$ (5)

Finally, the orbital speed can be found from equation 5:

Notice, that it is necessary to express r in units of meters.

$r = 95km \cdot \frac{1000m}{1km}$ ⇒ $95000m$

$v = \sqrt{\frac{(6.672x10^{-11}N.m^{2}/kg^{2})(4.4x10^{16}kg)}{95000m}}$

$v = 5.55m/s$

Hence, the orbital speed of Dactyl is $5.55m/s$