Finding Altitude When Given g = g0(Re/Re+A)^2

[Oannes]

Homework Statement

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g = g₀(R_e/(R_e + A))²

g is the acceleration due to gravity. g₀ is the acceleration of gravity at the surface of the earth, A is altitude, and R_e is the radius, approximately 6380 km. Assume g₀ = 9.8 meters per second squared. If the value of g is 9 meters per second squared, what is the Altitude in units of miles?

Homework Equations

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g = g₀(R_e/(R_e + A))²

The Attempt at a Solution

First, I plugged what I knew into the given equation:

9m/sec² = 9.8m/sec²(6380km/(6380km +A))²

Second, I divided m/sec from the right:

(9m/sec²)/(9.8m/sec²) = 0.9583
0.9583 = (6380/(6380 + A))²

Third, I took the square root of both sides:

0.9583 = 6380km/(6380km + A)

Fourth, I multiplied both sides by the denominator:

(6380km + A)(.9583) = (6380km/(6380km + A))(6380km + A)
6113.954km + .9583A = 6380km

Finally, I subtracted, divided, and converted units:

.9583A = 266.046km
A = 277.6229km

277.6229km *(.621 mi/ 1km) = 172.404 miles

My Questions: So while typing this problem up I started to understood more of what it is I needed to do and it became much easier. However, I still have some questions about converting the units and so forth.First, during the second step where I divided m/sec on the right side from the left side I canceled the units out during division as if they were a variable, I am pretty sure this is legal but I want to make sure that I am right in this assumption, otherwise I would have to convert m/sec^2 to km and I have no idea where to start on that one. Second, is taking the square root of both sides to get rid of the exponent a viable option? Once again I feel like this is true, but I am very paranoid and cautious when it comes to these things. Thanks in advance for any help given

 

[NascentOxygen]

First, during the second step where I divided m/sec on the right side from the left side I canceled the units out during division as if they were a variable, I am pretty sure this is legal but I want to make sure that I am right in this assumption,

Yes, that is what you need to do. It gives you an answer that is a dimensionless^❇ ratio

Second, is taking the square root of both sides to get rid of the exponent a viable option?

Yes, take the square-root of everything under a single big square-root sign, for each side.

^❇ this means it has no units