Inter-conversion of K.E. and P.E.

Kinetic energy (K.E.) and potential energy (P.E.) are two forms of energy that are inter-convertible. The conversion of K.E. to P.E. and P.E. to K.E. is governed by the principle of conservation of energy, which states that energy cannot be created or destroyed, only transferred or converted from one form to another.

When an object is lifted against the force of gravity, work is done on the object and potential energy is stored in the object. The potential energy is directly proportional to the height the object is lifted to, and is given by the formula:

P.E. = mgh

where m is the mass of the object, g is the acceleration due to gravity, and h is the height the object is lifted to.

When the object is released, the potential energy is converted to kinetic energy as the object falls under the influence of gravity. The kinetic energy is given by the formula:

K.E. = (1/2)mv^2

where m is the mass of the object, and v is the velocity of the object.

If the object is dropped from rest at a height h, its initial potential energy is given by:

P.E. = mgh

As the object falls, its potential energy decreases while its kinetic energy increases. At any given point during the fall, the sum of the potential and kinetic energies of the object is equal to its initial potential energy. This is expressed by the principle of conservation of energy:

P.E. + K.E. = constant

As the object falls, its velocity increases and its potential energy decreases. At the bottom of the fall, all of the potential energy has been converted to kinetic energy, and the velocity of the object is given by:

K.E. = (1/2)mv^2 = mgh

v^2 = 2gh

v = √(2gh)

Conversely, if an object is initially moving with a velocity v, its kinetic energy is given by:

K.E. = (1/2)mv^2

As the object is brought to a stop against the force of gravity, its kinetic energy decreases and its potential energy increases. At any given point during the rise, the sum of the kinetic and potential energies of the object is equal to its initial kinetic energy. This is expressed by the principle of conservation of energy:

K.E. + P.E. = constant

As the object rises, its velocity decreases and its potential energy increases. At the top of the rise, all of the kinetic energy has been converted to potential energy, and the height of the object above its initial position is given by:

P.E. = mgh

h = K.E. / mg

h = (1/2) v^2 / g

Thus, kinetic energy and potential energy are inter-convertible, and the total energy of the system remains constant.