**Conservation of Energy at Maximum Height of a Thrown Body**

There appears kinetic energy in a body due to motion and potential energy due to position. In a moving body both kinetic and potential energy can exist. For example, a flying aeroplane or a stone thrown above have both kinetic energy and potential energy. Then total energy of the body means the summation of kinetic and potential energy. So, total energy –

**E _{T} = E_{K} + E_{P}**

Kinetic energy of a body can be transformed into potential energy or potential energy can be transformed into kinetic energy. Many examples of this type can be cited. Now we will apply principle of conservation of energy of maximum height on a body thrown above.

Let a stone of mass in is thrown vertically above with a velocity v_{0} [Figure]. If the ground is considered as the reference level, then initial potential energy of the stone = 0 and initial kinetic energy = ½ mv_{0}^{2}. As the stone moves up its potential energy increases and velocity continues to decrease. So, during ascending kinetic energy of the stone is converted into potential energy. At height h if the velocity of the stone is v, (v < v_{0}), then kinetic energy of the stone at that point = ½ mv^{2} and potential energy = mgh. Hence, total energy of the stone = ½ mv^{2} + mgh. When the stone reaches to the highest position its velocity momentarily becomes zero. Then kinetic energy becomes zero and potential energy maximum. If the maximum height of the stone is H, then potential energy of the stone at that position = mgh.

So, at the highest position total kinetic energy of the stone is converted into potential energy.

After reaching the maximum height the stone again starts descending. Then opposite phenomenon occurs; potential energy of the stone then starts decreasing and its kinetic energy starts increasing. Just on the reference level only kinetic energy exists and potential energy becomes zero.

In this case it can be proved easily that if there is no dissipative force like frictional force, then total energy of the stone in the initial state (which is totally kinetic energy) is equal to total energy in the highest position of the stone (which is totally potential energy). That means, ½ mv_{0}^{2} = mgh

In other positions total energy remains unchanged. So,

**½ mv _{0}^{2} = mgH = ½ mv^{2} + mgh**

This principle is also applicable for a freely falling body. From the initial position from where the body was thrown abuse with velocity v_{0}, when the body again comes back to that initial position, the velocity becomes v_{0}. This time the total energy is kinetic. So, its total energy becomes ½ mv_{0}^{2}. Hence, thrown body obeys the principle of conservation of energy at the highest position.