A stone is projected vertically upward to reach maximum height h. The ratio of its kinetic energy to its potential energy at a height  will be5 : 44 : 51 : 44 : 1

Let's analyze the given scenario step by step:

1. The stone is projected vertically upward, which means it is thrown directly upwards.

2. As the stone moves upward, it gains potential energy due to its increasing height. Potential energy is given by the formula PE = mgh, where m is the mass of the stone, g is the acceleration due to gravity, and h is the height.

3. At the maximum height, the stone momentarily stops moving upward and starts to fall back down due to the force of gravity.

4. As the stone falls back down, it loses potential energy and gains kinetic energy. Kinetic energy is given by the formula KE = (1/2)mv^2, where m is the mass of the stone and v is its velocity.

5. The ratio of kinetic energy to potential energy at the maximum height can be calculated by dividing the kinetic energy by the potential energy.

6. The given ratio is 5:4, which means the kinetic energy is 5 units and the potential energy is 4 units.

7. Since the ratio of kinetic energy to potential energy is 5:4, we can write the equation: KE/PE = 5/4.

8. Substituting the formulas for kinetic energy and potential energy, we get: (1/2)mv^2 / mgh = 5/4.

9. Simplifying the equation, we can cancel out the mass of the stone: (1/2)v^2 / gh = 5/4.

10. Rearranging the equation, we get: v^2 / gh = 10/4.

11. Multiplying both sides of the equation by gh, we get: v^2 = (10/4)gh.

12. Taking the square root of both sides of the equation, we get: v = √((10/4)gh).

13. Therefore, the velocity of the stone at the maximum height is given by v = √((10/4)gh).

Please note that this analysis assumes ideal conditions and neglects factors such as air resistance.