The table shows information about four objects resting at the top of a hill.
Four Objects at Rest on a Hill
| Object | Mass $(kg)$ | Potential energy $(J)$ |
|---|---|---|
| W | 50 | 980 |
| X | 35 | 1,029 |
| Y | 62 | 1,519 |
| Z | 24 | 1,176 |
Which object is on the tallest hill?
Answer (2)
Z
Explanation
Problem Analysis We are given the mass and potential energy of four objects at the top of a hill and asked to determine which object is on the tallest hill.
Solution Plan The potential energy ( PE ) of an object is given by the formula PE = m g h , where m is the mass, g is the acceleration due to gravity, and h is the height. Since all objects are on the same planet, g is constant. Therefore, the height h is directly proportional to m PE . To find which object is on the tallest hill, we need to calculate m PE for each object and compare the values. The object with the largest m PE value is on the tallest hill.
Calculations for Object W For object W, the ratio m PE = 50 980 = 19.6 .
Calculations for Object X For object X, the ratio m PE = 35 1029 = 29.4 .
Calculations for Object Y For object Y, the ratio m PE = 62 1519 = 24.5 .
Calculations for Object Z For object Z, the ratio m PE = 24 1176 = 49.0 .
Comparison and Conclusion Comparing the ratios, we have: Object W: 19.6 Object X: 29.4 Object Y: 24.5 Object Z: 49.0 Since object Z has the largest ratio (49.0), it is on the tallest hill.
Examples
Understanding potential energy is crucial in various real-world scenarios. For instance, when designing roller coasters, engineers use the concept of potential energy to calculate the height needed for a thrilling drop. The higher the initial height, the greater the potential energy, which converts into kinetic energy as the coaster descends, providing the speed and excitement. Similarly, in hydroelectric power plants, the height of the water stored behind a dam determines the potential energy available to generate electricity. The greater the height, the more potential energy, and thus, the more electricity can be produced. This principle is also vital in understanding the stability of structures and the movement of objects in physics.
To find which object is on the tallest hill, we calculate the ratio of potential energy to mass for each object. Object Z has the highest ratio, 49.0, which indicates it is on the tallest hill. Thus, the answer is Object Z.
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