# The Naked Scientists Forum

### Author Topic: How do I calculate escape velocity at various elevations?  (Read 2193 times)

#### Agent Smith

• First timers
• Posts: 2
##### How do I calculate escape velocity at various elevations?
« on: 29/04/2010 18:22:12 »
[?]
How do I calculate the reduction in required escape velocity at various elevations?  Or, what is the advantage of increased elevation to distance traveled for a model rocket.

Assumption(s):
I have a model rocket that weighs 1kg (including engine) that is launched from the surface of the earth (assume radius 6371km) and the engine is able to lift it to 100m.

Question:
What is the formula to calculate the distance traveled if the same rocket is launched from an elevation 20km above the surface (i.e. 6391km).  I'd like to make a spreadsheet formula to do the calculation for various rocket weights, engines and various elevations.

Assume no atmosphere, or let's not worry about friction loss at this point in time.

Agent Smith

Mod edit - subject formatted as a question - please do this to help keep the forum tidy and easy to navigate.  Thanks.
« Last Edit: 30/04/2010 09:00:23 by BenV »

#### Agent Smith

• First timers
• Posts: 2
##### Re: How do I calculate escape velocity at various elevations?
« Reply #1 on: 29/04/2010 18:45:50 »
I think I found it:

Quote
Equation

The gravitational escape velocity equation is:

ve = √(2GM/R)

where

* ve is the escape velocity in kilometers/second (km/s)
* G is the Universal Gravitational Constant = 6.67*10−20 km3/kg-s2
* M is the mass of the planet or sun in kilograms (kg)
* R is the distance from the center of mass of the planet or sun to the center of the object in kilometers (km)
* √(2GM/R) is the square root of the quantity (2GM/R)

Important note: In previous Universal Gravitation Equations, G was stated in N-m2/kg2 and R in meters (m). However, it is more convenient to define escape velocity in kilometers/second (km/s). Thus, G is defined in km3/kg-s2 and R in km.

So given two radii of 6371 and 6391, I get escape velocities of 11.181 km/s and 11.163 km/s, respectively.

Doesn't seem to be much of an advantage to launch from a 20+ km elevation.

Next I need to calculate the increase in distance for the 1-kg rocket.

Thanks for listening,
Agent Smith

#### tommya300

• Hero Member
• Posts: 655
##### How do I calculate escape velocity at various elevations?
« Reply #2 on: 20/05/2010 06:24:25 »
I think I found it:

Quote
Equation

The gravitational escape velocity equation is:

ve = √(2GM/R)

where

* ve is the escape velocity in kilometers/second (km/s)
* G is the Universal Gravitational Constant = 6.67*10−20 km3/kg-s2
* M is the mass of the planet or sun in kilograms (kg)
* R is the distance from the center of mass of the planet or sun to the center of the object in kilometers (km)
* √(2GM/R) is the square root of the quantity (2GM/R)

Important note: In previous Universal Gravitation Equations, G was stated in N-m2/kg2 and R in meters (m). However, it is more convenient to define escape velocity in kilometers/second (km/s). Thus, G is defined in km3/kg-s2 and R in km.

So given two radii of 6371 and 6391, I get escape velocities of 11.181 km/s and 11.163 km/s, respectively.

Doesn't seem to be much of an advantage to launch from a 20+ km elevation.

Next I need to calculate the increase in distance for the 1-kg rocket.

Thanks for listening,
Agent Smith

Nice

Hey Smity
What duration of burn time will the rocket need to successfully escape 6371 to 6391 2R units. Then how many 1 Kg  engines will be needed  at 100 m per engine and how many stages will be needed since putting the engines side by side only would increase velocity and not distance. how do you compensate for the added masses of the each added engines since there is a handycap of weight per engine.

Would you need more than 64^64 engines and 64^64 stages for the engines. I guessed at this, but is it safe to say it will be a pyrimid affect. An intercontiental ballistic cannon? We need an angle to get the max distance and it needs to be larger than 45 deg lets be safe and give it 55 degress split it between going high where a free falling hyperbolic would be less affected by gravity. Fun thinking these things.

Melted my mind at one enhale.
« Last Edit: 20/05/2010 06:46:03 by tommya300 »

#### The Naked Scientists Forum

##### How do I calculate escape velocity at various elevations?
« Reply #2 on: 20/05/2010 06:24:25 »