Energy in elliptical orbits

Etot = total energy

a = semimajor axis

12.18 - Escape speed

Escape speed: The minimum speed required for

an object to escape a planet’s gravitational

attraction.

You know that if you throw a ball up in the air, the gravitational pull of the Earth will

cause it to fall back down. If by some superhuman burst of strength you were able to

hurl the ball up fast enough, it could have enough speed that the force of Earth’s gravity

would never be able to slow it down enough to cause it to return to the Earth.

The speed required to accomplish this feat is called the escape speed. Space agencies

frequently fire rockets with sufficient speed to escape the Earth’s gravity as they explore

space.

Given enough speed, a rocket can even escape the Sun’s gravitational influence,

allowing it to explore outside our solar system. As an example, the Pioneer 10

spacecraft, launched in 1972, was nearly 8 billion miles away from the Earth by 2003,

and is projected by NASA to continue to coast silently through deep space into the

interstellar reaches indefinitely.

At the right is an equation for calculating the escape speed from a planet of mass M. As

the example problem shows, the escape speed for the Earth is about 11,200 m/s, a little

more than 40,000 km/h. The escape speed does not depend on the mass of the object

being launched. However, the energy given to the object to make it escape is a function

of its mass, since the object’s kinetic energy is proportional to its mass.

The rotation of the Earth is used to assist in the gaining of escape speed. The Earth’s

rotation means that a rocket will have tangential speed (except at the poles, an unlikely

launch site for other reasons as well). The tangential speed equals the product of the

Earth’s angular velocity and the distance from the Earth’s axis of rotation.

An object will have a greater tangential speed near the equator because there the

distance from the Earth’s axis of rotation is greatest. The United States launches its

rockets from as close to the equator as is convenient: southern Florida. The rotation of

the Earth supplies an initial speed of 1469 km/h (408 m/s) to a rocket fired east from Cape Canaveral, about 4% of the required escape speed.

Derivation. We will derive the escape speed equation by considering a rocket launched from a planet of mass M with initial speed v. The

rocket, pulled by the planet’s gravity, slows as it rises. Its launch speed is just large enough that it never starts falling back toward the planet;

instead, its speed approaches zero as it approaches an infinite distance from the planet. If the initial speed is just a little less, the rocket will

eventually fall back toward the planet. If the speed is greater than or equal to the escape speed, the rocket will never return.

Escape speed

Minimum speed to escape planet’s

gravitational attraction

Some escape speeds

(^238) Copyright 2000-2007 Kinetic Books Co. Chapter 12