The escape velocity from Earth is about 11.186 km/s (6.951 mi/s; 40,270 km/h; 36,700 ft/s; 25,020 mph; 21,744 kn) at the surface. More generally, escape velocity is the speed at which the sum of an object's
For a spherically symmetric, massive body such as a star, or planet, the escape velocity for that body, at a given distance, is calculated by the formula
where G is the universal
When given a speed greater than the escape speed the object will asymptotically approach the hyperbolic excess speed satisfying the equation:
In these equations atmospheric friction (
The existence of escape velocity is a consequence of
For a given
The simplest way of deriving the formula for escape velocity is to use conservation of energy. For the sake of simplicity, unless stated otherwise, we assume that an object is attempting to escape from a uniform spherical planet by moving away from it and that the only significant force acting on the moving object is the planet's gravity. In its initial state, i, imagine that a spaceship of mass m is at a distance r from the center of mass of the planet, whose mass is M. Its initial speed is equal to its escape velocity, . At its final state, f, it will be an infinite distance away from the planet, and its speed will be negligibly small and assumed to be 0.
Kƒ = 0 because final velocity is zero, and Ugƒ = 0 because its final distance is infinity, so
where μ is the
Defined a little more formally, "escape velocity" is the initial speed required to go from an initial point in a gravitational potential field to infinity and end at infinity with a residual speed of zero, without any additional acceleration. All speeds and velocities measured with respect to the field. Additionally, the escape velocity at a point in space is equal to the speed that an object would have if it started at rest from an infinite distance and was pulled by gravity to that point.
In common usage, the initial point is on the surface of a
The escape velocity is independent of the mass of the escaping object. It does not matter if the mass is 1 kg or 1,000 kg; what differs is the amount of energy required. For an object of mass the energy required to escape the Earth's gravitational field is GMm / r, a function of the object's mass (where r is the radius of the Earth, G is the