A stone projected vertically upwards from the ground reaches a maximum height h. When it is at a height 3h/4, the ratio of its kinetic and potential energies is

The correct option is 1:3.

Let the mass of the stone be _m and the acceleration due to gravity be _g.
When the stone is projected vertically upwards, it reaches a maximum height $h$. At this maximum height, its velocity is zero, meaning its kinetic energy is zero, and its energy is entirely potential energy.
Therefore, the total mechanical energy ($E$) of the stone is equal to its potential energy at the maximum height:
$E=mgh$

According to the law of conservation of energy, the total mechanical energy remains constant throughout the motion.
Let us find the potential energy ($U_1$) and kinetic energy ($K_1$) when the stone is at a height of $\frac{3h}{4}$.

The potential energy at height $\frac{3h}{4}$ is:
$U_1=mg\left(\frac{3h}{4}\right)=\frac{3}{4}mgh$

Since the total energy is the sum of kinetic and potential energies:
$E=K_1+U_1$
We can find the kinetic energy ($K_1$) at this height by subtracting the potential energy from the total energy:
$K_1=E-U_1$
$K_1=mgh-\frac{3}{4}mgh=\frac{1}{4}mgh$

Now, we need to find the ratio of its kinetic energy to potential energy at this height:
$\frac{K_1}{U_1}=\frac{\frac{1}{4}mgh}{\frac{3}{4}mgh}=\frac{1}{3}$

Thus, the ratio of kinetic energy to potential energy at the height $\frac{3h}{4}$ is 1:3.