Escape Speed

Escape speed or escape velocity is the speed required for an accessible object that is required to escape the gravitational pull of its planet. It is usually considered ideal since it does not account for friction in the atmosphere. While the term “escape velocity” is often used, it is more appropriately described as a speed than a velocity due to its directionlessness; the escape speed grows in direct proportion to the mass of the original body and decreases with distance from it. Calculating the object’s escape velocity at a certain distance considers that it will slow down owing to the gravity of the large mass in the absence of extra acceleration. Even if it does, it will never fully come to a halt.

What is Escape Speed?

The following formula is used to compute the escape velocity of a spherically symmetric primary body with mass M situated at a distance d from its centre:

ve = 2GMd

Where G denotes the gravitational constant (G ≈ 6.67×10−11 m3·kg−1·s−2) on a global scale, the escape speed is independent of the fleeing item’s mass.The escape velocity of the Earth, or the speed at which it leaves the planet, is about 11.186 kilometres per second (40,270 kilometres per hour; 25,020 miles per hour; 36,700 feet per second).

Given a starting speed V more significant than the escape speed ve, the object will asymptotically approach the hyperbolic excess speed v∞, satisfying the equation:

v2= V2-ve2

In these equations, atmospheric friction (air drag) is not considered.

Derivation of Escape speed formula

The concept of energy conservation gives a simple method for calculating escape velocity (for another way, based on work, see below). We suppose that an object will move away from a uniform spherical planet to escape its gravitational field. That gravity is the only significant force acting on the moving item. 

Assume that a spaceship of mass m is positioned r from the planet’s centre of mass and that its initial speed matches its escape velocity. It will be an infinite distance away from the planet in its ultimate condition, and its speed will be negligible. We shall just examine kinetic energy K and gravitational potential energy Ug (ignoring air drag), therefore under the energy conservation equation, 

(K+Ug)i=(K+Ug)f

Where,

K= ½ mv2

U = GMm/r 

Here Ugf is zero as the distance is infinity, Kf will be zero as final velocity will be zero. Thus, we get:

½ mve2−GMm/r = 0 

½ mve2= GMm/r 

⇒ ve =√(2GM/r)

The minimum velocity required to escape from the gravitational influence of a massive body is given by:

ve = √(2gr)

Where,

g = GM/r2

The escape speed of the earth at the surface is approximately 11.186 km/s. That means “an object should have a minimum of 11.186 km/s initial velocity to escape from earth’s gravity and fly to infinite space.”

Typically, the starting point is on the surface of a planet or moon. The escape velocity on Earth’s surface is roughly 11.2 km/s, approximately 33 times the speed of sound. At 9,000 km altitude, it is equal to 7.1 km/s. It should be noted that this escape velocity is relative to a non-rotating frame of reference, not to a planet’s or moon’s moving surface.

Escape Speed in various scenarios

The direction in which the escaping body travels determines the escape velocity of a spinning body relative to its surface. Because the Earth’s rotational velocity at the equator is 465 m/s, a space shuttle, when ejected tangentially from the equator to the east, needs an initial velocity of about 10.735 km/s relative to the moving surface at the point of launch to achieve escape speed. In contrast, a shuttle ejected tangentially from the equator towards the west needs an initial velocity of 11.665 km/s relative to that moving surface to escape. 

Because surface velocity decreases with the cosine of geographic latitude, space launch facilities like the American Cape Canaveral and the French Guiana Space Centre are typically located as close to the equator as feasible.

Most of the time, achieving escape velocity almost instantly is impractical due to the acceleration implied, as well as the fact that if an atmosphere exists, the hypersonic speeds involved (on Earth, 11.2 km/s, or 40,320 km/h) will cause most objects to burn up due to aerodynamic heating or be torn apart by atmospheric drag. 

A spacecraft will accelerate steadily out of the atmosphere on a proper escape orbit until it reaches the escape velocity appropriate for its height (which will be less than on the surface). In many circumstances, the spacecraft will be placed in a parking orbit (such as a low Earth orbit of 160–2,000 km) and then accelerated to the escape velocity at that altitude, which will be slightly lower (about 11.0 km/s for a 200 km low Earth orbit).

Conclusion

Its track or trajectory is twisted when an object reaches escape speed but is not guided straight away from the planet. Even though it lacks a closed shape, this journey can be called an orbit. Assuming that gravity is the only significant force in the system, the object’s speed at any point along the trajectory will be equal to the escape velocity at that point because, due to energy conservation, the object’s total energy must always be zero, implying that it has escape velocity at all times. 

The path will be parabola-shaped, focusing close to the planet’s centre of mass. A genuine escape requires avoiding collisions with the planet or its atmosphere, which would cause the object to crash. This path is adopted when travelling away from the source.