Missing question:
"Determine (a) the astronaut’s orbital speed v and (b) the period of the orbit"
Solution
part a) The center of the orbit of the third astronaut is located at the center of the moon. This means that the radius of the orbit is the sum of the Moon's radius r0 and the altitude ([tex]h=430 km=4.3 \cdot 10^5 m[/tex]) of the orbit:
[tex]r= r_0 + h=1.7 \cdot 10^6 m + 4.3 \cdot 10^5 m=2.13 \cdot 10^6 m[/tex]
This is a circular motion, where the centripetal acceleration is equal to the gravitational acceleration g at this altitude. The problem says that at this altitude, [tex]g=1.08 m/s^2[/tex]. So we can write
[tex]g=a_c= \frac{v^2}{r} [/tex]
where [tex]a_c[/tex] is the centripetal acceleration and v is the speed of the astronaut. Re-arranging it we can find v:
[tex]v= \sqrt{g r}= \sqrt{(1.08 m/s^2)(2.13 \cdot 10^6 m)}=1517 m/s = 1.52 km/s [/tex]
part b) The orbit has a circumference of [tex]2 \pi r[/tex], and the astronaut is covering it at a speed equal to v. Therefore, the period of the orbit is
[tex]T= \frac{2 \pi r}{v} = \frac{2\pi (2.13 \cdot 10^6 m)}{1517 m/s} =8818 s = 2.45 h[/tex]
So, the period of the orbit is 2.45 hours.
