Just a quick question of the escape velocity equation

In summary, The escape velocity of a body can be calculated using the equation √(2GM)/R, where G is the gravitational constant, M is the mass of the body, and R is the radius of the body. This means that the speed needed to escape Earth's gravity at a distance of 1 mile above the surface would be 11185.08934 m/s. However, this does not take into account the atmosphere, which would be a factor at such a low altitude. The escape velocity is the minimum speed required for a body to leave the gravitational field without any additional propulsion. Rockets, which have continuous propulsion, can escape at lower speeds.
  • #1
zeromodz
246
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Okay, I just found this new equation (new to me) for the escape velocity of a body.

escape velocity = √(2GM)/R

So say for instance I plug this into the escape velocity of the earth

I am going to say the radius is 4001 miles so I it would take a mile above the Earth to escape (4001 * 1600) = 6401600 meters
escape velocity = √((2)(6.674*10^-11)(6.0*10^24)/(6401600)
= 11185.08934 m/s

This obviously isn't the speed you need to escape the Earth's gravity a mile off of its surface because we have left Earth already with much lower speeds. This is so fast we would get a little time dilation, can someone tell me what I am doing wrong?

Thanks in advanced.
 
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  • #2
The Apollo missions acheived speeds around 11000 m/s or more, so you're number seems OK, other than the atmosphere would be an issue if only 1 mile up.
 
  • #3
Wait I am confused, the escape velocity it the speed you need to leave the Earth's system, Is this data I collected implying I must go that speed to go above the Earth for one mile?

That doesn't make sense, I wouldn't have to go that speed to go just a mile above the surface.
 
  • #4
zeromodz said:
I am going to say the radius is 4001 miles so I it would take a mile above the Earth to escape

What does this even mean? I can't make heads or tails of this business you're talking about "a mile above to escape." Also, time dilation (indeed, all relativistic effects) are negligible for the vast majority of travel up until about .1c (for example, at this speed gamma is 1.00000000067).
 
  • #5
11 000 m/s is the speed you need to escape from Earth if you start a mile up. It's not the speed needed to travel 1 mile up from the ground.

BTW, the escape velocity is the speed needed for a FREEFALLING object to escape. That means it doesn't apply to rockets, as rockets have propulsion. To illustrate, you could fire your rockets and inch upwards at 1 mm/s and still escape Earth completely.
 
  • #6
A helicopter can go a mile above Earth's surface, so obvosly escape velocity is not how fast you need to go to go to a certain height.

Escape velocity is the speed at which you must go to escape Earth's gravity. For instance, let's say you shoot a rocket straight up at 10000m/s. It will fly up and up and up, but will constantly be slowed down by Earth's gravity, until it slows all the way to 0 and starts falling back down...and it will hit the ground. Below escape velocity, "what goes up must come down" (kind of).

If however, you shoot the thing at 12000m/s, it will also fly up and up and up. Earth's gravity will still slow it, but keep in mind Earth's gravity diminishes as you go further away. If you're going at escape velocity, your rocket will basically never be slowed down to 0. It might approach 0, but it will keep getting further away from earth, and Earth's gravity will keep diminishing, with the result that it will keep on going into infinity and never fall back down.

Also, if you were hanging out at a far-far-far distance from Earth at a speed of 0, and you let Earth's gravity pull you in, you will end up traveling at escape velocity when you finally hit the ground (disregarding the atmosphere).
 
  • #7
Keep in mind that escape velocity is the speed (it's actually a scalar) that a body would need to have to escape the gravitational field it is in, without the benefit of any propulsion.

So the reason why the number is so large, is because that is the speed you would need to initially have, to be able to then "coast" (no propulsion!) out of the gravitational field. The reason rockets seem to escape at lower speeds is because they are continually applying force.
 
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FAQ: Just a quick question of the escape velocity equation

1. What is the escape velocity equation?

The escape velocity equation is a mathematical formula that calculates the minimum speed an object needs to escape the gravitational pull of a larger object, such as a planet or moon. It is expressed as v = √(2GM/R), where v is the escape velocity, G is the gravitational constant, M is the mass of the larger object, and R is the distance between the two objects.

2. How is the escape velocity equation used?

The escape velocity equation is used to determine the speed at which an object must travel in order to break free from the gravitational pull of a larger object. This can be useful in space travel, as it helps scientists and engineers determine the necessary velocity for spacecraft to escape the Earth's orbit and travel to other planets or moons.

3. What factors affect the escape velocity?

The escape velocity is affected by two main factors: the mass of the larger object and the distance between the two objects. The larger the mass of the object, the greater the escape velocity will be. Similarly, the closer the two objects are, the higher the escape velocity will be.

4. Can the escape velocity equation be used for any object?

No, the escape velocity equation can only be used for objects that have a spherical shape and a constant mass. This means it cannot be used for irregularly shaped objects or objects with varying mass, such as asteroids or comets.

5. Are there any limitations to the escape velocity equation?

Yes, the escape velocity equation has some limitations. It assumes a vacuum environment with no air resistance, and it does not take into account any external forces, such as the gravitational pull of other objects. Additionally, it only calculates the minimum speed needed to escape the gravitational pull and does not take into account the direction of the velocity.

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