Question

A body is falling from a height $\mathrm{h}$. After it has fallen from a height $\frac{\mathrm{h}}{2}$, it will possess

A

only potential energy

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B

only kinetic energy

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C

half potential and half kinetic energy

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D

more kinetic and less potential energy

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Solution

The correct option is C half potential and half kinetic energyStep 1. Given dataIt is given that a body is falling from a height $\mathrm{h}$. After it has fallen from a height $\frac{\mathrm{h}}{2}$.Step 2. Formula to be usedThe energy possessed by the body due to virtue of its motion is called Kinetic energy.The kinetic energy is, $\mathrm{K}.\mathrm{E}.=\frac{1}{2}{\mathrm{mv}}^{2}$Here, $\mathrm{m}$ is mass, $\mathrm{v}$ is velocity and $\mathrm{K}.\mathrm{E}.$ is kinetic energy.The energy possessed by a body due to the virtue of its position or configuration is called potential energy. $\mathrm{P}.\mathrm{E}.=\mathrm{mgh}$Here, $\mathrm{m}$ is mass, $\mathrm{g}$ is gravitational field strength and $\mathrm{h}$ is height.Step 3. Find the energy which possesses by the given height Let a body of mass ′m′ placed at a height $\prime \mathrm{h}\prime$ above the ground, and start falling down from rest which is shown in above figure.When the body falls from a height $\prime \mathrm{h}\prime$ to $\frac{\mathrm{h}}{2}$, the potential energy is given as $\frac{\mathrm{mgh}}{2}$.The body covers the distance $\frac{\mathrm{h}}{2}$ with a velocity $\mathrm{v}$. We make use of the third equation of motion to obtain the velocity of the body.According to the third equation of motion is,${\mathrm{v}}^{2}-{\mathrm{u}}^{2}=2\mathrm{as}$Here, $\mathrm{v}$ is the velocity at any point in time, $\mathrm{u}$ is the initial velocity, $\mathrm{a}$ is the acceleration and $\mathrm{s}$ is the distance travelled. From the given, $\mathrm{u}=0$ $\mathrm{a}=\mathrm{g}$ $\mathrm{s}=\mathrm{x}$So, ${\mathrm{v}}^{2}-0=\frac{2\mathrm{gh}}{2}$ ${\mathrm{v}}^{2}=\mathrm{gh}$Now, Kinetic energy $=\frac{1}{2}{\mathrm{mv}}^{2}$ $=\frac{\mathrm{mgh}}{2}$So, at height $\frac{\mathrm{h}}{2}$ the energy possessed is half potential energy and half the kinetic energy.Hence, option $\mathrm{C}$ is the correct answer.

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