# Specific Mechanical Energy (1)

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epsilon =
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MichaelBartmess.Specific Mechanical Energy (1)
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This Specific Mechanical Energy equation computes the mechanical energy per unit mass ) as a constant characterizing a circular satellite orbit.  This equation uses the Earth Gravitational Constant, mu, and the input length of the orbital radius of the satellite orbit, r.

INSTRUCTIONS: Choose the preferred units and enter the following:

• (a) This is the radius of the satellite's orbit

Specific Mechanical Energy of the Satellite (ε): The calculator returns the energy per unit mass of a satellite in a circular orbit of the specified radius. The result is in units of Grays, where  1 Gy = 1 "Joule"/"kilogram" = 1 "meter"^2/"sec"^2

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#### The Math / Science

ε = -μ/(2*r)

where:

##### Derivation

The mechanical energy of an object is a sum of its kinetic energy (K) and potential energy (U):

(1) E = K + U

The kinetic energy of satellite can be written using the equation:

(2) K = ½⋅m⋅v², where m is the satellite's mass and v is the satellite's velocity

And the potential energy of a satellite can be written using the equation:

(3) U = -(G*M*m)/r, where G is the universal gravitational constant, M is the mass of the Earth,

Therefore, substituting (2) and (3) into equation (1), we get:

(4) E = 1/2 * m * v^2 - (G*M*m)/r

In a circular orbit, Newton's 2nd Law dictates:

(5) (G*M*m)/r^2 = (m*v^2)/r

And multiplying both sides of (5) by r we get:

(6) (G*M*m)/r = (m*v^2)

Using (6) and substituting for (m*v^2)  (4) becomes:

(7) E = 1/2 * m * v^2 - (G*M*m)/r  = 1/2 * (G*M*m)/r - (G*M*m)/r = -(G/2)*(M*m)/r

Setting mu to be the Earth's gravitational constant:

(8) mu = G*M

We get from (7):

(9) E = -mu/(2*r) * m

And by definition the specific mechanical energy of the satellite is the mechanical energy per unit mass:

(10) epsilon = E/m = (-mu/(2*r) * m)/m

And finally we get:

(11) epsilon = -mu/(2*r)