What Velocity Is Needed to Escape Earth's Gravity?

[zephyrous]

Homework Statement

A question I made up myself:
With what velocity would an object need to be launched at to totally overcome gravity of the Earth (to have a constant velocity out in space somewhere.) Neglecting other planets, atmosphere friction, etc.

Homework Equations

Newton's Second Law
Newton's Universal law of gravitation

The Attempt at a Solution

I figured mass would be negligable if I combined N2L and NGrav and said:

F=ma=G*m*(earth's mass)*(r^-2)

So acceleration is -G(mass earth)(r(t)^-2)

But the thing is that I want to launch this baby so fast that "R" is a function of time. So when I integrate over a time differential I run into problems, because I want to integrate from t=0 to t="t" so what pops out for velocity is g(mass earth)/(r(t))- g(mass earth)/(earth's radius) plus initial velocity if you will.

I don't even know if this is correct, and I know if I take the next step to get to position I will get into natural logarithms and I'd really not trust myself then. I guess whenever I get to my position equation then that becomes my "r(t)" and I just have myself a differential equation of some sort.

I'm sure this problem has been done before and I'm almost certain it takes differential equation even at the most simple level. Does anyone know of a basic explanation of how to calculate this that only uses relatively simple undergraduate calculus and physics? I know it's connected to the topics of trajectory motion but I'm guessing the solution is more complicated than the typical "throw the ball in the air how high will it go?" brand of problems.

Thanks

 

[diazona]

Are you familiar with potential energy? The problem is trivially easy to do if you use energy conservation - no calculus required. For inspiration, take a look at the Wikipedia article on escape velocity.

Alternatively, you can do it by solving the differential equation you obtained. It's a little more complicated that way, but it's an interesting exercise (and it's necessary if you want to know how much time it takes the object to reach a certain height).

 

[zephyrous]

I figured it out!
I guess it's a concept called escape velocity and it can be derived a few different ways.
Very Cool!

 

[zephyrous]

Are you familiar with potential energy? The problem is trivially easy to do if you use energy conservation - no calculus required. For inspiration, take a look at the Wikipedia article on escape velocity.

Alternatively, you can do it by solving the differential equation you obtained. It's a little more complicated that way, but it's an interesting exercise (and it's necessary if you want to know how much time it takes the object to reach a certain height).

Yep! very cool

 

[rwisz]

The question you have posed is a complex one, and the solution involves a combination of calculus and physics principles. To completely overcome the gravity of the Earth, an object would need to be launched at a velocity that allows it to escape the Earth's gravitational pull and continue moving out into space without being pulled back by gravity. This velocity is known as the escape velocity and can be calculated using the following equation:

Ve = √(2GM/r)

Where Ve is the escape velocity, G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth to the object's starting point. This equation takes into account the Earth's gravity and the object's initial distance from the Earth's center.

To understand this equation, we can break it down into its components. The term "GM" represents the combined gravitational force of the Earth and the object. The term "r" represents the distance between the Earth's center and the object's starting point. By taking the square root of this value, we are able to find the velocity needed to overcome this gravitational force and escape the Earth's pull.

In terms of calculus, this equation can also be thought of as the integral of the gravitational force over the distance traveled by the object. As the object moves away from the Earth, the force of gravity decreases, but the distance traveled increases. By integrating over this distance, we are able to find the total amount of energy needed to escape the Earth's gravitational pull.

In summary, the velocity needed to completely overcome the Earth's gravity is known as the escape velocity and can be calculated using a combination of physics and calculus principles. It is an important concept in understanding the dynamics of objects in space and is used in spacecraft launches and other space missions.