Answered

The table shows information about four objects resting at the top of a hill.



\begin{table}

\captionsetup{labelformat=empty}

\caption{Four Objects at Rest on a Hill}


Object		Mass (kg)Potential energy (J)

W		50980

X		351,029

Y		621,519

Z		241,176




\end{table}



Which object is on the tallest hill?

AW
BX
CY
DZ

Verified Answer :

  • The gravitational potential energy formula \(PE = mgh\) helps determine height as \(h = \frac{PE}{mg}\).
  • We compare the heights by calculating the ratio \(\frac{PE}{m}\) for each object.
  • The calculated ratios are: W: \(19.6\), X: \(29.4\), Y: \(24.5\), Z: \(49.0\).
  • Since object Z has the highest ratio, it rests on the tallest hill, \(\boxed{Z}\).

Explanation

  1. Analysis of Potential Energy and Height

The gravitational potential energy (\(PE\)) of an object of mass \(m\) resting at a height \(h\) is defined by the formula:
\[PE = mgh\]
where \(g\) is the acceleration due to gravity, which is a constant. We are given the mass and potential energy for four objects (W, X, Y, and Z) and want to find which object is on the tallest hill. To compare the heights of the hills (\(h\)), we can rearrange the formula to solve for \(h\):
\[h = \frac{PE}{mg}\]
Since \(g\) is the same for all objects, comparing the heights is equivalent to comparing the ratio \(\frac{PE}{m}\). A higher ratio indicates a greater height \(h\).

  1. Calculating Height Ratios

We calculate the ratio \(\frac{PE}{m}\) for each object based on the provided table:

  • For Object W: \(\frac{980}{50} = 19.6\)
  • For Object X: \(\frac{1029}{35} = 29.4\)
  • For Object Y: \(\frac{1519}{62} = 24.5\)
  • For Object Z: \(\frac{1176}{24} = 49.0\)
  1. Conclusion

Comparing the calculated ratios, we see the values are \(19.6, 29.4, 24.5,\) and \(49.0\). The object Z, with a ratio of \(49.0\), has the largest value, meaning it is resting on the tallest hill. Therefore, the object on the tallest hill is Z. The final answer is \(\boxed{Z}\).

Examples

This type of problem relates to how physics formulas describe real-world relationships. Imagine designing a roller coaster where you need to ensure a car reaches a certain height to have enough energy to complete a loop. By rearranging the potential energy formula \(PE = mgh\), engineers can determine the required height based on the mass of the train and the energy needed, ensuring safety and performance. This is a fundamental concept in civil engineering and physics.

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