How the calculated total velocity derived ?

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In summary: December 2018 (EST)In summary, the equation for calculating the total delta-v needed for launch is: VEH^2 + VE^2= (deltaV)^2
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Venus50
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From the article "Which Way To Mars?. Trajectory Analysis. by Kelsey B. Lynn" that I have read, it states that the Hohmann Trajectory at earth,V1=32.76km/s and the Earth velocity,VE=29.82km/s,so the hyperbolic excess velocity,VH=[V1-VE]=2.940km/s.All these parameters lead to the calculation of the total velocity needed for launch from earth[know as the delta V].To launch from one of the earth'spoles,the delta V is a combination of VH[hyperbolic excess velocity] and VES[escape velocity from earth],that is delta V=[VH^2+VES^2]^.5 =11.6km/s. The value of VES=11.18KM/S,of course.What I would like to know is how the formula{deltaV=[VH^2+VES^2]^.5} is derived? Any help for explanation would be appreciated. Thanks a lot.
 
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  • #2
they are derived from the energy equation.

The energy of a satellite can be described using the following formula:

E = 1/2 V^2 - mu/r

For the situation on the surface of the Earth, with a velocity deltaV (on the pole the rocket doesn't get extra velocity due to the Earth's rotation), the energy is:

E_1 = 1/2 (deltaV)^2 -mu/Re

For a parabolic orbit (the satellite is always at the escape velocity), the velocity is zero at infinite distance. When you actually want to go somewhere, you want to have that excess velocity VEH. At r = infinity (you have escaped), the energy is then:

E_2 = 1/2 VEH^2

Energy is conserved after giving the delta-V (assuming the delta-V was there immediately). Equating the two equations gives:

1/2 VEH^2 = 1/2 (deltaV)^2 - mu/Re

it happens to be the case that escape velocity is written as:
VE = sqrt(2*mu/r)
so: mu/Re = 1/2 VE^2
at r=Re

so, finally:

1/2 VEH^2 = 1/2 (deltaV)^2 -1/2 VE^2

or:
VEH^2 + VE^2= (deltaV)^2

et voila
 
  • #4
Thanks,remcook. i also see that of Hellfire now.
 
  • #5
Velocity vector components from orbital elements

Venus50 said:
From the article "Which Way To Mars?. Trajectory Analysis. by Kelsey B. Lynn" that I have read, it states that the Hohmann Trajectory at earth,V1=32.76km/s and the Earth velocity,VE=29.82km/s,so the hyperbolic excess velocity,VH=[V1-VE]=2.940km/s.All these parameters lead to the calculation of the total velocity needed for launch from earth[know as the delta V].To launch from one of the earth'spoles,the delta V is a combination of VH[hyperbolic excess velocity] and VES[escape velocity from earth],that is delta V=[VH^2+VES^2]^.5 =11.6km/s. The value of VES=11.18KM/S,of course.What I would like to know is how the formula{deltaV=[VH^2+VES^2]^.5} is derived? Any help for explanation would be appreciated. Thanks a lot.
Note that...

a : semimajor axis of the orbit
e : eccentricity of the orbit

1 astronomical unit = 1.49597870691E+11 meters

GMsun = 1.32712440018E+20 m^3 sec^-2


The canonical velocity in a hyperbolic orbit, in general, is found from

Vx’’’ = -(a/r) { GMsun / a }^0.5 sinh u

Vy’’’ = +(a/r) { GMsun / a }^0.5 (e^2 - 1)^0.5 cosh u

Vz’’’ = 0

Where u is the eccentric anomaly and r is the current distance from the sun.


The canonical velocity in an elliptical orbit, in general, is found from

Vx’’’ = -sin Q { GMsun / [ a (1-e^2) ] }^0.5

Vy’’’ = (e + cos Q) { GMsun / [ a (1-e^2) ] }^0.5

Vz’’’ = 0

Where Q is the true anomaly.

The triple-primed vectors would be rotated (negatively) by the angular elements of the orbit (w,i,L) to heliocentric ecliptic coordinates.


Rotation by the argument of the perihelion, w.

Vx'' = Vx''' cos w - Vy''' sin w

Vy'' = Vx''' sin w + Vy''' cos w

Vz'' = Vz''' = 0


Rotation by the inclination, i.

Vx' = Vx''

Vy' = Vy'' cos i

Vz' = Vy'' sin i


Rotation by the longitude of ascending node, L.

Vx = Vx' cos L - Vy' sin L

Vy = Vx' sin L + Vy' cos L

Vz = Vz'


The unprimed vector [Vx, Vy, Vz] is the velocity in the orbit, referred to heliocentric ecliptic coordinates.

Jerry Abbott
 
Last edited:

1. What is total velocity and why is it important in calculations?

Total velocity is the combined velocity of an object, taking into account both its speed and direction of motion. It is important in calculations because it helps us understand how fast and in what direction an object is moving, which is crucial in many scientific experiments and real-world applications.

2. How is total velocity calculated?

Total velocity is calculated by adding together the individual components of an object's velocity, typically represented by its speed and direction in a specific coordinate system. This can be done mathematically using vector operations or graphically using vector diagrams.

3. What factors can affect the calculated total velocity?

The calculated total velocity can be affected by various factors, such as changes in speed or direction of an object, the presence of external forces, and errors in measurements. It can also be influenced by the choice of coordinate system and the accuracy of the calculations.

4. How is total velocity used in real-world scenarios?

Total velocity is used in many real-world scenarios, such as predicting the path of a projectile, analyzing the motion of a moving vehicle, or studying the behavior of fluids in pipes. It is also crucial in fields such as astrophysics, where total velocity is used to understand the motion of celestial bodies.

5. Can total velocity be negative?

Yes, total velocity can be negative. This means that the object is moving in the opposite direction of the chosen coordinate system. For example, if a car is moving south, its total velocity would be negative if the chosen coordinate system has north as the positive direction.

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