Question

An object in space has altitude of 2210 km, velocity of 7000 m/s and flight path angle of 20 degrees. Find the eccentricity, semi-major axis, angular momentum/mass, and kinetic and potential energies/mass

Answer 1

Eccentricity = 0.0557

Semi-major axis = 9,095 km

Angular momentum/mass = 60,116
`(km^(2) )/(Sec)`

Kinetic energies/mass = 24,500 KJ

Potential energies/mass = 21,673 KJ

Step-by-step explanation:

Eccentricity

To find the eccentricity use the following formula

Eccentricity = [ Altitude from the earth x (
`Velocity^(2)` / Gravitational parameter for the Earth ) ] - 1

Where

Altitude from the earth radius = Radius of the earth + Altitude of the earth from radius = 6,378 km + 2,210 km = 8,588 km

Velocity = 7,000 m/s

Gravitational parameter for the Earth = 3.986004418 ×
`10^(14)`

Eccentricity = ?

Placing values in the formula

Eccentricity = [ ( 8,588 km x 1000 ) x (
`7000^(2)` / 3.986004418 ×
`10^(14)` ) ] - 1

Eccentricity = 0.0557

Semi-major axis

Total Distance = Semi-major axis x ( 1 - Eccentricity )

Where

Total Distance = 8,588 km

Eccentricity = 0.0557

8,588 x 1,000 m = Semi-major axis x ( 1 - 0.0557 )

8,588,000 m = Semi-major axis x 0.9443

Semi-major axis = 8,588,000 m / 0.9443

Semi-major axis = 9,094,567.40 m

Semi-major axis = 9,094.56740 km

Semi-major axis = 9,095 km

Angular momentum/mass

L = MVR

L/M = VR

Where

V = Velocity =
`(7,000 m/s)/(1000)` = 7 km/s

R = Total Radius = Radius of Earth + Altitude = 6,378 km + 2,210 km = 8,588 km

Placing values in the formula

L/M = 7 km/s x 8,588 km = 60,116
`(km^(2) )/(Sec)`

Kinetic energies/mass

Ke = I
`W^(2)`

Ke =
`(1)/(2) mr^(2)`
`W^(2)` ( Where I =
`mr^(2)` )

`(ke)/(m)` =
`(r^2)/(2)`
`W^(2)`

`(ke)/(m)` =
`(r^2)/(2)`
`(V^2)/(r^2)` (
`W^(2)` =
`(V^2)/(r^2)` )

`(ke)/(m)` =
`(V^2)/(2)`

`(ke)/(m)` =
`(7000^2)/(2)`

`(ke)/(m)` = 24,500,000 J

`(ke)/(m)` = 24,500 KJ

Potential energies/mass

PE = mgh

`(PE)/(m)` = gh

Where

g = 9.807
`(m)/(s^2)`

h = 2,210 km x 1,000 = 2,210,000 m

Placing values in the formula

`(PE)/(m)` = 9.807
`(m)/(s^2)` x 2,210,000 m

`(PE)/(m)` = 21,673,470 J

`(PE)/(m)` = 21,673.470 KJ

`(PE)/(m)` = 21,673 KJ