Get equation for velocity of a satellite

[rcrx]

Derive an equation for the velocity around Earth. Satellite has no mass. Radius not given. Only period, G, and ME (earth)

G = 6.67X10^-11 nm^2/kg^2 | Earth = 5.98x10^24 kg | Period = T = 1.20x10^4s

I know I can find radius, and then find velocity. But I am supposed to derive an equation to suit the given information.

Homework Equations

V = √GM/r
V = 2πr / T

The Attempt at a Solution

I'm not positive about my ability to cut r from the equations and merge them?

 

[Rake-MC]

You say that you know that you can solve for radius and then find velocity, but you're supposed to derive an equation to suit the given information.

Well, solving for radius and then velocity is in a way doing what they asked, except just don't solve for r numerically!

Re arrange one of the equations so it's r = [...] and then put the [...] in as r for the other equation.

 

[baba89]

I would approach this problem by first defining the variables and their units. G is the universal gravitational constant with units of m^3/kg*s^2, ME is the mass of the Earth with units of kg, and T is the period of the satellite with units of s.

We can begin by using the equation for the velocity of a satellite in circular orbit, which is V = √GM/r. We know that the satellite has no mass, so we can remove the variable for mass from the equation, leaving us with V = √GM/r.

Next, we can use the given information to solve for r. The period of the satellite, T, is equal to 1.20x10^4s. We also know that the distance traveled in one orbit is equal to the circumference of the circle, which is 2πr. Therefore, we can set up an equation:

T = 2πr/V

Solving for r, we get r = VT/2π.

Now, we can substitute this value for r into the first equation:

V = √GM/(VT/2π)

Simplifying, we get V = √2πGM/T

We can then substitute in the values for G and ME:

V = √(2π)(6.67x10^-11 m^3/kg*s^2)(5.98x10^24 kg)/1.20x10^4s

Simplifying further, we get V = √1.19x10^14 m^2/s^2

Therefore, the final equation for the velocity of a satellite with no mass in orbit around Earth is V = 3.45x10^7 m/s.

In summary, by using the equations for velocity of a satellite and the circumference of a circle, we were able to derive an equation for the velocity of a satellite around Earth using the given information.