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Definition/Summary
The minimum launch speed needed to ensure a projectile on the surface of a body will completely break free from its gravitational pull.
Escape velocity (being a speed, rather than a velocity) is a scalar.
Escape velocity is the same for any mass of projectile, and for any direction of launch. If the direction of launch is not radial (vertical), the projectile will follow a parabola.
Equations
[tex]v_{escape} = \sqrt{\frac{2GM}{r}}[/tex]
Extended explanation
"Projectile":
A projectile is something which moves without any force being applied during its journey, except for an initial impulsive force, or launch.
Space rockets do not leave the Earth as projectiles: their rockets fire continuously (until they reach the desired orbit).
A projectile is something you "hit and forget".
Conservation of energy:
Escape speed (ignoring air resistance, rotation of the body, and the presence of any other bodies) is the speed needed to achieve zero speed "at infinity", and can be calculated using conservation of (mechanical) energy:
[tex]KE\ =\ \frac{1}{2}mv^2\ \ \ PE\ =\ -\frac{GmM}{r}[/tex]
[tex]KE(r)\ -\ KE({\infty})\ =\ PE({\infty})\ -\ PE(r)[/tex]
[tex]\frac{1}{2}mv_{escape}^2\ -\ 0\ =\ 0\ -\ \left(-\frac{GmM}{r}\right)[/tex]
and so:
[tex]v_{escape}\ =\ \sqrt{\frac{2GM}{r}}[/tex]
where m is the mass of the projectile, M is the mass of the planet, r is the radius of the planet, and G is the universal gravitational constant.
g, the gravitational constant, or "force of gravity", on the surface of the body, is [itex]GM/r^2[/itex]
From a rotating body:
On the surface of a body which is rotating, a projectile already has the velocity of the surface, and so, relative to the surface, the escape velocity may be slightly more or less than the figure given above, and will depend on the direction of launch (for a vertical launch, it will always be less, except at the poles). The difference will be greatest at the equator, and zero at the poles.
"Velocity"
Velocity, in scientific English, means a speed and a direction. But in ordinary English, velocity and speed have the same meaning. In "escape velocity", the ordinary meaning has triumphed.
A similar confusion arises with g-force, which in scientific English is an acceleration, not a force.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
The minimum launch speed needed to ensure a projectile on the surface of a body will completely break free from its gravitational pull.
Escape velocity (being a speed, rather than a velocity) is a scalar.
Escape velocity is the same for any mass of projectile, and for any direction of launch. If the direction of launch is not radial (vertical), the projectile will follow a parabola.
Equations
[tex]v_{escape} = \sqrt{\frac{2GM}{r}}[/tex]
Extended explanation
"Projectile":
A projectile is something which moves without any force being applied during its journey, except for an initial impulsive force, or launch.
Space rockets do not leave the Earth as projectiles: their rockets fire continuously (until they reach the desired orbit).
A projectile is something you "hit and forget".
Conservation of energy:
Escape speed (ignoring air resistance, rotation of the body, and the presence of any other bodies) is the speed needed to achieve zero speed "at infinity", and can be calculated using conservation of (mechanical) energy:
[tex]KE\ =\ \frac{1}{2}mv^2\ \ \ PE\ =\ -\frac{GmM}{r}[/tex]
[tex]KE(r)\ -\ KE({\infty})\ =\ PE({\infty})\ -\ PE(r)[/tex]
[tex]\frac{1}{2}mv_{escape}^2\ -\ 0\ =\ 0\ -\ \left(-\frac{GmM}{r}\right)[/tex]
and so:
[tex]v_{escape}\ =\ \sqrt{\frac{2GM}{r}}[/tex]
where m is the mass of the projectile, M is the mass of the planet, r is the radius of the planet, and G is the universal gravitational constant.
g, the gravitational constant, or "force of gravity", on the surface of the body, is [itex]GM/r^2[/itex]
From a rotating body:
On the surface of a body which is rotating, a projectile already has the velocity of the surface, and so, relative to the surface, the escape velocity may be slightly more or less than the figure given above, and will depend on the direction of launch (for a vertical launch, it will always be less, except at the poles). The difference will be greatest at the equator, and zero at the poles.
"Velocity"
Velocity, in scientific English, means a speed and a direction. But in ordinary English, velocity and speed have the same meaning. In "escape velocity", the ordinary meaning has triumphed.
A similar confusion arises with g-force, which in scientific English is an acceleration, not a force.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!