# Trouble understanding the idea of escape velocity

1. Jan 14, 2014

### subzero0137

I'm having trouble understanding the idea of escape velocity. How can an object escape the gravity of a massive object like the Earth? No matter what the velocity the object is, doesn't Newton's law of gravity imply that eventually, the force of gravity will cause the object to decelerate, and then fall back to earth?

2. Jan 14, 2014

### dextercioby

Not trully so, it may happen that for some initial conditions, the conic solution of the Kepler problem is a parabola or hyperbola.

3. Jan 14, 2014

### subzero0137

Hmm...I haven't come across the "conic solution of the Kepler problem" before, so I'll have to look that up or wait for my lecturer to get to that part. But could you explain why I can't simply use the Newton's law of gravity, which states that $F=\frac{GMm}{r^2}$, and argue that no matter how fast the object travels away from Earth, it will always experience a deceleration and thus eventually fall back to Earth?

4. Jan 14, 2014

### dextercioby

The law you wrote is the scalar version, but forces are vectors. The acceleration vector (the force vector if you prefer) is not collinear to the velocity one for motion along a curve. The 2-body problem involves motion along a curved trajectory (a conic section).

5. Jan 14, 2014

### subzero0137

I see. But what if the trajectory of the object is not a curve? What if a rocket simply lifted off with the escape velocity radially away from the Earth?

6. Jan 14, 2014

### dextercioby

The Newtonian potential is conservative. Even for the motion you propose, you're free to increase the KE of the rocket as much as you like, which from a velocity value upwords would be higher than the Potential energy which the rocket has at the surface of the earth. The rocket would then evade the gravitational field of the earth and would not orbit it.

7. Jan 14, 2014

### subzero0137

Why would that happen? Sorry if I'm being difficult, but I just don't understand it!

8. Jan 14, 2014

### Staff: Mentor

No matter which direction you start from, the further away from earth you get, the closer to directly away from it your trajectory points.

9. Jan 14, 2014

### SteamKing

Staff Emeritus
10. Jan 14, 2014

### subzero0137

So basically, we can't use Newton's law of gravitation on its own. We have to use it alongside the conservation of energy law, so when the total mechanical energy is 0, the object pretty much travels at a constant speed, despite being in a force field?

11. Jan 14, 2014

### SteamKing

Staff Emeritus
Who said anything about the object traveling at constant speed? When the velocity of the object makes its kinetic energy equal to the gravitational potential energy, then the object is said to have reached escape velocity. Objects can't achieve any velocity, let alone escape velocity, without accelerating from rest.

12. Jan 14, 2014

### HallsofIvy

Staff Emeritus
If an object leaves the earth, or any other massive body, the "escape velocity" is the initial velocity ("escape velocity" is defined assuming no additional force so the velocity is steadily decreasing) needed that the velocity will stay non-zero to "infinity".

13. Jan 14, 2014

### subzero0137

I see. So an object that has 'escaped' the gravity of a massive object will still decelerate. I understand that bit. But in the wiki article, it says that the velocity will drop to 0 at infinity, and not stay non-zero. Am I missing something?

14. Jan 15, 2014

### jbriggs444

For practical purposes, this "at infinity" can be taken as shorthand for "far enough away that it doesn't mater" and this 0 can be taken as "close enough to zero that it doesn't matter". Realistically, there is, no such thing as actually being at infinity.

More formally, an object at escape velocity has enough velocity so that no matter how far you want it to go, if you wait long enough, it will get that far. And for any finite distance, you will not have to wait infinitely long.

If you go far enough in mathematics you'll find things like the extended reals or non-standard analysis where infinities [or infinitesimals] can be put on a firm enough footing so that it can be formally meaningful to talk about something being zero "at infinity". Until then, it would be wise to avoid naive reasoning about infinity as if it were just another place or another number.

15. Jan 15, 2014

### ASG141

unless there is another field, you cannot escape the earth.

if you go too far off, you enter the mars or venus sphere of influence or may be zones where the sun's field is peaking.

mapping the space G field wil give you an idea of where an object will be directed at a given (x,y,z,t).

16. Jan 15, 2014

### D H

Staff Emeritus
You never reach infinite distance, so your velocity is always non-zero. It is however tending toward zero, and becomes zero in the limit r→∞.

That's the case for a parabolic trajectory. For a hyperbolic trajectory, the velocity tends toward some non-zero quantity, typically called v.

17. Jan 15, 2014

### ASG141

the parabolic as per galileo is a general form for linear field L >>> O object with horizontal velocity x and vertical velocity y. the earth's field is considered to be a line of infinite length and constant G.

at greater distance, the earth's field becomes a point source with decreasing G and spherical density.

18. Feb 13, 2014

### mpresic

I think I see the confusion. Newton's law of gravity tells us the force of gravity diminishes as the inverse square of the distance. That is doubling the distance from the source of gravity (center of the Earth) allows the gravity force to decrease to one-fourth it's former strength. The gravity is weakening. Now if the "rocket"
is given a sufficient initial velocity, the rate at which it covers distance (velocity) is greater than the rate at which the gravity field is weakening (as a result of the increased distance). Consequently, the rocket is allowed to escape

19. Feb 13, 2014

### snorkack

Newton´s original laws do not express energy conservation, actually. Vis viva does come out, but it was almost two centuries after Newton that energy conservation came out.

But converging infinite series were known to Newton. So above escape speed, an object is indeed always in gravity field, and always slowing slightly, but never will stop nor slow beyond a certain nonzero speed.

20. Feb 13, 2014

### dauto

The OP's mistake seems to be the belief that if something is constantly decelerating then its speed must eventually reach zero. That's just not true. (This is kind of a reversed Zeno paradox. In the classic paradox the distance between Achilles and the tortoise is mistakenly believed never to reach zero because it consists of an infinite number of steps. The OP thinks the speed must reach zero because of the infinite number of steps taken reducing the speed by the continuous action of gravity. They are both wrong)

Last edited: Feb 13, 2014