While attempting a landing on the moon, astronauts had to change their landing site and land at a spot that was 4 kilometers away from the original site. assuming that they were at a height of 137 meters, calculate the horizontal velocity of the spacecraft during touchdown if it lands in a free-fall mode without using retro engines. consider gravity = 1.63 meters/second^2.

a) 143.23 meters/second

b) 233.33 meters/second

c) 308.88 meters/second

d) 333.44 meters/second

The closest answer would be C.
The choices given do not give the exact value. 

To answer this, you just need to remember the main formula:



Where:
d = distance/displacement
g = acceleration due to gravity
t = time in flight
Vi = initial velocity.

With this formula, you derive all the formulas you need to look for certain components. You need to keep in mind of the following:

---if you are looking for a vertical component(y), you need to use values of vertical motion.



*Viy is always 0m/s at the beginning of a free-fall.



                                             

---If you are looking for a horizontal component(x), you need to use values of horizontal motion. 



*g is always 0m/s² when taking horizontal motion into account. 



                                                       
--- time is the only value that is both vertical and horizontal. 

Okay, let's get back to solving your problem. Let's see what your given is first:
dy = 137m (as long as it refers to height, it is vertical distance)
dx = 4km (the word, far or away usually indicates horizontal distance)
g = 1.63m/s²

The question is how fast was it going horizontally and we can derive it from our equation:

 

We use this because x means horizontal. But notice that we do not have time yet. So how are we going to solve this 2 variables missing? The key is that time is a horizontal and vertical component. Whatever time it took moving horizontally, it is the same vertically as well. So we use the vertical formula to derive time:




                                           

Now plug in what you know and solve for what you don't know:






The total time in flight is 12.965s.
Let's round it off to 13s. 

Now that we know that, we can use this in the horizontal formula:



Hold up! Look at the unit of the horizontal distance. It is in km but all our units are expressed in m so we need to convert that first.

1km = 1,000m
4km = 4,000m

Our new horizontal distance is 4,000m.

Okay, let's wrap this up by solving for what is asked for, using all the derived values. 





The horizontal velocity is 308m/s. 

Such a long explanation I know, but hopefully, you learned from it. 

This did i unseated, sorry