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# Derive quantitatively and apply the concepts of gravitational force and gravitational potential energy in radial gravitational fields to a variety of situations

– The concept of escape velocity ?esc=√2???

**Escape Velocity**: The initial velocity required by a projectile to rise vertically and just escape the gravitational field of a planet.- During the rise experienced when travelling at an escape velocity, the projectile’s kinetic energy transforms into gravitational potential energy such that:

Ek (initially) = Ep (finally)

- By considering the kinetic and gravitational energies of a projectile, it can be shown mathematically that the escape velocity of a planet depends only upon the universal gravitational constant, the mass of the planet, and the radius of the planet:

kinetic energy loss : , where v_{i } = initial velocity

potential energy gain : , where *r *is the radius of the earth

Escape velocity =

- The escape velocity does not depend upon any intrinsic property of the projectile.
- The escape velocity of Earth works out to be approximately 40 000 km h
^{-1}.

– total potential energy of a planet or satellite in its orbit U=−????

**Gravitational Potential Energy (E**The energy of a mass due to its position within a gravitational field._{p}):- As part of the Law of Universal Gravitation, the inverse square law relates the strength of a gravitational field and the distance from the centre of its source.
- Thus the force of gravitational attraction between an object and a planet will only drop to zero when the object is an infinite distance from the planet.
- At infinity (or a very large distance away) is the level of zero potential energy in space.
- Using this law we calculate force and using to calculate gain in potential energy.
- Thus, in space, gravitational potential energy is defined as the work done to move an object from infinity (or a very large distance away) to a point within a gravitational field.
- It can be shown mathematically that:

where:

- m
_{1}= mass of planet (kg). - m
_{2}= mass of object (kg). - r = distance separating masses (m).
- G = universal gravitational constant.

– total energy of a planet or satellite in its orbit U+K=−???2?

- to calculate this we will first calculate Kinetic energy

- potential energy:

– energy changes that occur when satellites move between orbits (ACSPH096)

Since Total Energy only depends on radial distance (as mass remains constant), as the orbit changes , the total energy of the satellite changes

Change in energy

To change orbit, rockets are fired which increase or decrease E_{k} , to accommodate for this energy. As soon as the satellite changes altitude transformation between E_{k} and E_{p} occur , as the satellite settles down in its new orbit.

Extract from *Physics Stage 6 Syllabus © 2017 *NSW Education Standards Authority (NESA)