A particle is released from height \(s\) above the surface of the earth. At a certain height its K.E is 3 times of P.E. The height from the surface of the earth and the speed of the particle at the instant are respectively:
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In problems involving energy conservation, use the relationship between K.E and P.E to find the velocity and height at certain points.
The total mechanical energy is defined as the sum of kinetic energy (K.E.) and potential energy (P.E.):
\[
E_{\text{total}} = \text{K.E} + \text{P.E}
\]
Since total mechanical energy is conserved, it remains constant throughout the particle's motion. At a specific point, it is given that the K.E. is three times the P.E.:
\[
\text{K.E} = 3 \cdot \text{P.E}
\]
Therefore, at this point, the total mechanical energy can be expressed as:
\[
E_{\text{total}} = 3 \cdot \text{P.E} + \text{P.E} = 4 \cdot \text{P.E}
\]
This indicates that at that height, the potential energy is \( \frac{1}{4} \) of the total mechanical energy. The particle's speed can then be determined using energy conservation principles.
Applying energy conservation and solving, the height from the surface is found to be \( \frac{s}{4} \), and the speed is \( \sqrt{\frac{3gs}{2}} \).
