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 :  kinetic energy loss,   where vi  = initial velocity

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

Escape velocity = 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 (Ep): The energy of a mass due to its position within a gravitational field.
  • 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 force and using gain in potential energy 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: gravitational potential energy

          where:

  • m1 = mass of planet (kg).
  • m2 = 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 kinetic energy
  • potential energy: gravitational potential energy  
  • total 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

  • Total Energy in orbit with radius r_1
  • Total Energy in orbit with radius r_2

Change in energy change in energy

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

 
 
 

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