Introduction:
When a stone is projected vertically up, it follows a parabolic trajectory due to the force of gravity acting on it. At the maximum height 'h', the stone momentarily comes to rest before falling back down. We need to find the ratio of the stone's kinetic energy to its potential energy at a height of 4h/5.

Understanding Kinetic and Potential Energies:
- Kinetic Energy (KE): It is the energy possessed by an object due to its motion. For a stone, its kinetic energy depends on its mass and velocity.
- Potential Energy (PE): It is the energy possessed by an object due to its position or height. For a stone, its potential energy depends on its mass, acceleration due to gravity, and height.

Calculating the Ratio:
To calculate the ratio of kinetic to potential energies, we need to find the respective energies at a height of 4h/5. Let's assume the mass of the stone is 'm'.

Step 1: Find the Kinetic Energy:
At a given height, the stone will have a certain velocity. We can use the conservation of energy to find this velocity.

Conservation of Energy:
At the maximum height (h), the stone has only potential energy and no kinetic energy. Therefore, we can equate the initial potential energy to the final potential energy at 4h/5.

Initial Potential Energy = Final Potential Energy
mgh = mgh'

Here, h' is the final height (4h/5).

Simplifying the equation, we get:
h = h'/5

Step 2: Find the Potential Energy:
At a height of 4h/5, we can calculate the potential energy using the formula:
PE = mgh'

Step 3: Find the Kinetic Energy:
To find the kinetic energy, we need to determine the velocity at a height of 4h/5. We can use the equation of motion:

Final Velocity^2 = Initial Velocity^2 + 2g(h - h')

Since the stone is projected vertically up, the initial velocity is positive and the final velocity is zero at the highest point (h).

0 = Initial Velocity^2 + 2g(h - h')

Simplifying the equation, we get:
Initial Velocity = sqrt(2gh')

Using the initial velocity, we can calculate the kinetic energy using the formula:
KE = (1/2)mv^2

Step 4: Calculate the Ratio:
Finally, we can calculate the ratio of kinetic to potential energies using the formulas obtained in Step 2 and Step 3:

Ratio = KE / PE
Ratio = [(1/2)mv^2] / (mgh')

Simplifying the equation, we get:
Ratio = v^2 / (2gh')

Conclusion:
The ratio of kinetic to potential energies at a height of 4h/5 is given by v^2 / (2gh'), where v is the initial velocity and h' is the final height. By following the steps mentioned above, we can calculate this ratio for a stone projected vertically up to reach a maximum height 'h'.

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