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SecretSnow

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In summary, the concept of escape speed is the minimum speed required for an object to reach infinity, with just enough kinetic energy for it to come to a rest at infinity. This means that the kinetic energy of the object will ultimately approach zero as it reaches infinity due to the universal nature of gravity. However, this does not mean that the object has zero speed at infinity, as it will still have a constant speed that approaches zero in the limit. Therefore, while the textbook may state that the kinetic energy is zero at infinity, this is referring to the total mechanical energy, which can be negative, positive, or zero depending on the trajectory of the object. The concept of escape velocity itself is not about actually escaping, but rather about meeting the definition

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SecretSnow

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Physics news on Phys.org

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Fightfish

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Recall that the gravitational force approaches zero as the object approaches infinity - so there is no contradiction.

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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.

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SecretSnow

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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?

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MrWarlock616

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tiny-tim

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SecretSnow said:The spaceship must have some speed when it escapes from Earth …

The spaceship

… when the spaceship approaches infinity, it will at least have a constant speed and hence kinetic energy right?

The spaceship

it is

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.

… the kinetic energy of the spaceship shouldn't be zero ultimately right? By the way, my textbook says its zero.

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SecretSnow

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tiny-tim said:Hi SecretSnow!

The spaceshipneverescapes from Earth …

howeverfar away it is, the gravity from Earth is alwaysgreaterthan zero.

The spaceshipneverhas constant speed …

it isalwaysslowing 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.

whichtextbook, and whatexactlydoes it say?

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.

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russ_watters

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tiny-tim

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SecretSnow said:… 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?

that's the

(so no need to prove anything)

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.

that's right, escape velocity isn't about

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?

right

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

and the quote?

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SecretSnow

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tadchem

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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.

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tiny-tim

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SecretSnow said:… there's really nothing called escape speed right? It'll get pulled back eventually right?

no, anything

anything

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HallsofIvy

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Yes, and it is

"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.

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SammyS

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The spaceship doesn't have zero K.E. at escape speed.SecretSnow said:Hi guys, how can a spaceship supposedly have zero Kinetic energy at escape speed?

The spaceship's K.E. at escape speed is: [itex]\displaystyle KE_\text{ escape}= (1/2)m(v_\text{escape})^2\ .[/itex]

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.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!

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SecretSnow

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Is there any images or animations to show this? :D

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tiny-tim

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as HallsofIvy says …

HallsofIvy said:"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 …

… so saying that escape velocity is when the total energy (potential energy plus kinetic energy) is zero is just a tautolgy

SecretSnow said:What's a hyperbolic orbit? In what way is it hyperbolic if the spaceship just flies off from Earth?

an

a

a

ie a parabolic orbit is the same as a trajectory with exactly escape velocity

At escape speed, the kinetic energy is equal to zero because all of the energy of the object is used to overcome the gravitational force of the planet or body it is trying to escape from. This means that the object has just enough energy to reach infinite distance from the planet, but not enough to continue moving away.

The escape speed can be calculated using the formula: v = √(2GM/r), where v is the escape speed, G is the gravitational constant, M is the mass of the planet or body, and r is the distance from the center of the planet or body to the object.

No, the escape speed varies depending on the mass and size of the planet or body. For example, the escape speed from Earth is about 11.2 km/s, while the escape speed from the Moon is only about 2.4 km/s.

No, an object cannot have kinetic energy at escape speed. As mentioned before, all of the energy is used to overcome the gravitational force, leaving no energy for movement.

The kinetic energy decreases as an object reaches escape speed because the gravitational potential energy increases. This means that more energy is being used to overcome the gravitational force, leaving less energy for movement.

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