# Why is KE = 0 at escape speed?

1. Jan 3, 2013

### SecretSnow

Hi guys, how can a spaceship supposedly have zero Kinetic energy at escape speed? If this is the case, then once it has reached the area to escape the gravitational attraction of earth (till it is very minimal) is it saying that it has zero speed? Wouldn't it start falling back because it is just exactly at the threshold of the "escape area"? Then the spaceship can be said not able to escape earth right? What I am trying to say is that if this is the case, the kinetic energy of the spaceship shouldn't be zero ultimately right? By the way, my textbook says its zero. I don't know why lol. Thanks a lot people!!!

2. Jan 3, 2013

### Fightfish

Escape speed is the minimum speed required for the object to just reach infinity. What this means is that the kinetic energy of the object during launch is just sufficient for the object to come to a rest as it approaches infinity. The "rest" part is why the object has zero kinetic energy at infinity - not during the launch!
Recall that the gravitational force approaches zero as the object approaches infinity - so there is no contradiction.

3. Jan 3, 2013

### D H

Staff Emeritus
You misunderstood or your text wasn't clear.

An object with non-zero velocity has non-zero kinetic energy.

The total mechanical energy of an object subject to the gravitational force of some central body is the sum of it's non-negative kinetic energy and non-positive gravitational energy. This sum can be negative (a bound orbit), positive (a hyperbolic trajectory), or zero (a parabolic trajectory). It's this final kind of trajectory, a parabolic one, that defines escape velocity. It's the total mechanical energy that is zero, not the kinetic energy.

4. Jan 3, 2013

### SecretSnow

The spaceship must have some speed when it escapes from earth so why at that moment the KE is zero? Even when there's no additional propulsion given, when the spaceship approaches infinity, it will at least have a constant speed and hence kinetic energy right? Since infinity cannot be reached, then it should never be at rest right?

Erm I don't get the part where you say the total mechanical energy is zero. Assuming that the initial kinetic energy is zero, and the gravitational potential energy is a certain negative value, and that mechanical energy is conserved, why would the final mechanical energy be zero?

5. Jan 3, 2013

### MrWarlock616

The kinetic energy supplied to an object with the escape velocity is only just enough to help it clear the gravitational field of the earth. The velocity of the upward going object will slowly decrease because of the gravitational pull and finally become zero.

6. Jan 3, 2013

### tiny-tim

Hi SecretSnow!
The spaceship never escapes from earth …

however far away it is, the gravity from earth is always greater than zero.
The spaceship never has constant speed …

it is always slowing down …

its speed approaches a constant in the limit as its distance approaches infinity.

If the initial speed was escape velocity, then that constant is zero.

which textbook, and what exactly does it say?

7. Jan 3, 2013

### SecretSnow

Lol you guys are kind people :D

Anyway, how do you know that the constant speed reached (when the spaceship eventually goes so far that gravity's pull is negligible) is zero if the initial velocity is escape speed? Is there a proof for this? And if gravity pull is universal, then there's really no escape speed (neglecting the presence of other planets) since any object can never escape from Earth. If this is the case, then again KE will never be zero since there's always a net force acting on and there's always acceleration. And hence there's always kinetic energy right?

I'm using University Physics 13th edition. lol.

8. Jan 3, 2013

### Staff: Mentor

It is a limit. Have you had calculus yet? As distance increases toward infinity, speed APPROACHES zero.

9. Jan 3, 2013

### tiny-tim

that's the definition of escape velocity

(so no need to prove anything)

that's right, escape velocity isn't about actually escaping, it's just a name for the speed that meets the definition
right
and the quote?

10. Jan 3, 2013

### SecretSnow

Hi guys well I just figured everything out. It does eventually approach zero because like throwing a ball up to a certain distance, it slows down to 0 eventually, as gravity is universal. In that sense, there's really nothing called escape speed right? It'll get pulled back eventually right? Then how can one derive the escape velocity when there is none in the first place?

11. Jan 3, 2013

The spaceship has both potential energy (due to its position in the earth's gravitational field) and kinetic energy (due to its speed).
If the potential energy (negative) is greater in magnitude that its kinetic energy (positive), then the Total Energy is negative and the spaceship is trapped in the earth's gravitational field (in either an orbit or a collision course).
If the potential energy (negative) is lesser in magnitude that its kinetic energy (positive), then the Total Energy is positive and the spaceship will escape the earth's gravitational field (in a hyperbolic orbit).
When the Total Energy is zero, the spaceship will continue climbing out of the earth's gravitational field on a parabolic path, losing speed until eventually (at the Restaurant at the End of the Universe, find itself parked infinitely far from Earth.

12. Jan 3, 2013

### tiny-tim

no, anything less than escape speed will be pulled back eventually

anything greater than escape speed will eventually move at just above a constant non-zero speed

13. Jan 3, 2013

### HallsofIvy

Yes, and it is total energy, not kinetic energy that is 0 at "escape velocity".

"potential energy" is always relative to some arbitrary "0" point. For gravitational potential energy it is customary to choose the "0" point "at infinity" so that potential energy at any point close to the earth is negative. That means that if the total energy, kinetic energy plus the negative potential energy, is negative, the object will eventually fall back to the earth and if it is positive, it can go "infinitely far" and still have positive kinetic energy. "Escape speed" is the speed at which total energy, kinetic energy and the negative potential energy is exactly 0- the minimum energy necessary not to fall back to earth.

14. Jan 3, 2013

### SammyS

Staff Emeritus
The spaceship doesn't have zero K.E. at escape speed.

The spaceship's K.E. at escape speed is: $\displaystyle KE_\text{ escape}= (1/2)m(v_\text{escape})^2\ .$
If the spaceship is launched from earth at escape speed, then ignoring friction and gravitational forces from any other bodies, the spaceship's K.E. will approach zero as the spaceship's distance from earth approaches infinity.

15. Jan 4, 2013

### SecretSnow

What's a hyperbolic orbit? In what way is it hyperbolic if the spaceship just flies off from Earth?
Is there any images or animations to show this? :D

16. Jan 4, 2013