Concepts: satellites, escape speed and exercise practice

4.1 Escape speed

The escape speed is the smallest launch speed for which a body can go away from the Earth and never fall back. It is found by setting the total energy at the surface to zero, because the body just reaches infinity with zero speed. That gives the square root of twice G M over R, about 11.2 km per second for the Earth. The value does not depend on the direction of projection, only on the speed, and it does not depend on the mass of the body because the mass cancels.

For the Moon the escape speed is much smaller, about 2.4 km per second, which is one reason the Moon has no atmosphere: gas molecules could reach escape speed at lunar temperatures. For the Sun the escape speed from its surface is hundreds of kilometres per second, which is why the solar wind is held to the solar system.

The escape speed is larger than the orbital speed of a low satellite by the factor root two. A body launched at the orbital speed goes into a circular orbit around the Earth, while one launched at the escape speed leaves the Earth's influence entirely.

4.2 Satellites

A satellite in a circular orbit has the gravitational force as its centripetal force, so G M m over r squared equals m v squared over r. The orbital speed is the square root of G M over r, and the period is two pi r divided by that speed. Substituting the speed shows the period squared is proportional to r cubed, which is Kepler's third law again.

The total energy of a satellite in a circular orbit is negative, and its magnitude is the kinetic energy, while the potential energy is minus twice the kinetic energy. The kinetic energy is half the magnitude of the potential energy. This is the virial relation for a circular orbit and a useful check on energy calculations.

Geostationary satellites orbit above the equator with a period of one day, which fixes their radius at about 42,000 km from the Earth's centre, about 36,000 km above the surface. A polar satellite orbits over the poles at a lower altitude and can scan the whole surface as the Earth turns beneath it. Both are applications of the same orbital equations.

4.3 Weightlessness in an orbit

Inside an orbiting spacecraft every object falls with the same acceleration toward the Earth, so the object and the floor fall together and the normal reaction between them is zero. The astronaut floats not because gravity is absent but because gravity is the only force acting, and everything falls at the same rate. This is called weightlessness in free fall, and the same effect can be felt briefly in a falling lift.

A satellite is in continuous free fall toward the Earth, but its tangential speed carries it around the Earth at the same rate as it falls. That combination is what an orbit is. If the tangential speed were removed, the satellite would fall straight down; if the speed were larger than the escape speed, it would leave.

The concept is often examined as a statement question. The weight of an astronaut in orbit is not zero, but the apparent weight, which is the normal reaction from the floor, is zero. The distinction between the true weight and the apparent weight is the same as in the lift problems of the laws of motion chapter.

4.4 Exercise practice: the chapter exercises

Exercise 7.1 asks for the ratio of the gravitational force between the Earth and the Moon to that between the Sun and the Moon, and exercise 7.2 asks for the force between two spheres using the centre to centre separation. Exercise 7.3 verifies Kepler's third law for the Earth's orbit. Exercise 7.4 finds the acceleration due to gravity at a height equal to the radius of the Earth, using the ratio of distances squared.

Exercise 7.5 asks where the gravitational field of the Earth and the Moon cancels on the line joining them, and exercise 7.6 finds the gravitational potential and field at a point on the axis of a ring. Exercise 7.7 uses the potential energy of a system of three masses. Exercise 7.8 asks for the work done to move a mass from the surface to a given height using the full potential energy expression.

Exercises 7.9 and 7.10 find the escape speed from a planet and check the units. Exercise 7.11 and 7.12 use the period-radius relation for satellites, and exercise 7.13 finds the energy needed to move a satellite from one orbit to another. Exercise 7.14 finds the height of a geostationary satellite, and exercise 7.15 checks the weightlessness statement. Exercise 7.16 and 7.17 use the energy relations to find speeds at different points of an orbit.

A reliable method for all of these is to write the energy or force equation with the full distances R plus h, then take the limit h much smaller than R only if the question asks for an approximation. State the zero of potential energy at infinity in every energy answer.