When a stone is projected vertically, it moves against the force of gravity until it reaches its maximum height, h. As it ascends, its kinetic energy (KE) decreases, and its potential energy (PE) increases until it momentarily becomes zero at the peak.
After reaching a height of 4h, the ratio of kinetic energy to potential energy can be calculated. At maximum height, the potential energy is given by the equation:
PE = mgh, where:
- m = mass of the stone
- g = acceleration due to gravity
- h = height
At height 4h, the potential energy becomes:
PE = mg(4h) = 4mgh.
The kinetic energy at this point can be determined from the conservation of energy principles. At the moment it reaches height 4h, all the initial kinetic energy would have converted into potential energy minus any KE remaining. Given the context of the problem, let’s denote the initial kinetic energy right before reaching the height 4h as:
KE_initial = mgh (at height h) = 1mgh.
For the purposes of comparing kinetic and potential energy at height 4h, we can express KE at that point as:
KE = KE_initial – PE = mgh – 4mgh = -3mgh, revealing that KE at height 4h is now negative, indicating it has come to a complete stop prior to dropping back down.
Thus, when calculating the ratio of kinetic energy to potential energy at height 4h, we find:
Ratio = KE / PE = -3mgh / 4mgh = -3/4.
However, since the energy is conceptually represented in a ratio form as per your problem context, the key would be to simply express this ratio contextually as negative.
As such, if numerically referenced against the mentioned set of numbers like 1, 5, 4, 2, 4, 5, the outcomes of kinetic to potential remain significant. Ultimately, one must consider a clear numerical computation to answer precisely per your text format.
This showcases that the ratio is dependent on our defined kinetic energy compared to potential, emphasizing the energy transformations occurring during vertical projection.