An object of mass m = 50 g is released from rest from a height h = 150 m above the ground. Calculate the kinetic energy and the potential energy of the object at the height H = h/5.

 and Energy Conservation

Mechanical energy describes the ability of an object to do work. The work done on an object arises from a force applied over a distance (W=Fdd) which either accelerates the object thus changing its motional energy (kinetic energy), or stores energy by changing its position (potential energy). For instance, when a moving car is brought to rest, the work done by the frictional force on the tires is equal to the kinetic energy of the car, KE=1/2 mv2. In addition, forces which are provided by the car's engine can do work in climbing up a hill which is stored as gravitational potential energy, PE=mgh. The mechanical energy of an object is equal to the sum of the potential plus kinetic energies, i.e. E = PE + KE, and is a direct measure of the total energy available to an object as its speed and position changes from one point to another.

In special cases where energy is not lost to the environment due to nonconservative forces such as friction, the mechanical energy of a system of masses remains constant as the object or objects move in space. In other words, the energy of the system stays the same for all times in the life of the objects. Since there is no energy created or destroyed as the objects travel along on their paths, the forces which do work exchange energy between potential and kinetic according to the energy initially provided to the system. This is the basis for the law of  conservation of energy which can be written as E = PE + KE = constant. As an example, consider an apple of mass m which is initially a height h above the ground. When the apple is hanging from the tree, it stores an amount of gravitational potential energy as it is held motionless so that the initial mechanical energy is Etop = PEtop = mgh. If the apple is cut from the tree, forces due to gravity do work which cause it to fall and speed up at the same time. Just before it hits the ground, the potential energy is zero such that the mechanical energy arises solely due to the kinetic energy, Ebot = KEbot = 1/2 mv2. Using the conservation of energy for this frictionless problem, we can equate the mechanical energy at the top and bottom of the tree, or more specifically                                       E top =E top

PE top = KE bot

mgh = 1/2 mv2.

It is clear from the equations above that the mechanical energy allows you to relate the position of the apple to its velocity as it falls from the tree. As potential energy is lost from the decrease in height above the ground, kinetic energy is gained while the object speeds up.

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