Circular Motion and the Law of Gravitation -- question

[John Kim]

Homework Statement

Matt Damon is stuck on Mars. He needs to get o the planet and into orbit to rendezvous with the rescue team, which will be orbiting the planet at the same radius as Phobos, one of Mars’s moons. His goal is to determine what his take-of speed should be so that he makes it into orbit. Assume circular orbits throughout this problem. (i) First, Matt Damon throws a ball with an initial speed of v0 at an angle theta above the horizontal and measures that the ball lands a horizontal distance d away from where he threw it. Ignoring his height, determine the acceleration due to gravity on Mars, g, in terms of v0, d, and theta.

(ii) Next, he measures the period, T, of Phobos’s orbit as well as the radius of Mars, R (he does the latter using a method similar to that of Eratosthenes). Given this information and your result for g above, determine the radius r of orbit of Phobos, in terms of T,R,d, v0, and theta (and any constants). Hint: Think about Kepler’s third law. Use that the gravitational force must be a centripetal force.

(iii) Lastly, he recalls from his astronaut training that the orbital speed of the rescue team will be vf , and hence he needs to end up orbiting at radius r with speed vf . Determine the speed, v, with which he must launch his rocket to make it to the proper orbit (you may neglect Mars’s rotational motion). Your answer should be in terms of vf,d, v0, theta, T, and R.

Homework Equations

F=gm1m2/r^2
a = v^2/r
T=2piR/v

The Attempt at a Solution

https://imgur.com/a/f8Bv2
(here is the attempt from Chegg) however, i am not sure how we got part iii) when solving for the velocity he must launch the rocket. Really clueless on how he got v = sqrt(Vf^2 + 2gr) for the components of velocity

 

[haruspex]

v = sqrt(Vf^2 + 2gr) for the components of velocity

You mean, how they got that for the launch speed, right? The components would be Vf horizontal and 2gr vertical (though if r is being measured from the centre of Mars, shouldn't that be 2g(r-R)? It says r is the radius of the orbit, not the altitude.)
Can you see how they get those components?
That said, that result is only valid if the orbital altitude is small compared with R, which is not what the diagram suggests. Looks to me like an incorrect solution.