To determine which object is on the tallest hill, we can use the relationship between potential energy (PE), mass (m), gravitational acceleration (g), and height (h). The formula for potential energy is given by:
[tex]\[ \text{PE} = m \cdot g \cdot h \][/tex]
Given:
- [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex]
- Potential energy and mass for each object.
We need to solve for height ([tex]\( h \)[/tex]) for each object. The formula rearranges to:
[tex]\[ h = \frac{\text{PE}}{m \cdot g} \][/tex]
Let's calculate the height for each object:
1. Object W:
[tex]\[ m_W = 50 \, \text{kg}, \, \text{PE}_W = 980 \, \text{J} \][/tex]
[tex]\[ h_W = \frac{980}{50 \times 9.8} \][/tex]
[tex]\[ h_W = \frac{980}{490} \][/tex]
[tex]\[ h_W \approx 2.0 \, \text{m} \][/tex]
2. Object X:
[tex]\[ m_X = 35 \, \text{kg}, \, \text{PE}_X = 1,029 \, \text{J} \][/tex]
[tex]\[ h_X = \frac{1,029}{35 \times 9.8} \][/tex]
[tex]\[ h_X = \frac{1,029}{343} \][/tex]
[tex]\[ h_X \approx 3.0 \, \text{m} \][/tex]
3. Object Y:
[tex]\[ m_Y = 62 \, \text{kg}, \, \text{PE}_Y = 1,519 \, \text{J} \][/tex]
[tex]\[ h_Y = \frac{1,519}{62 \times 9.8} \][/tex]
[tex]\[ h_Y = \frac{1,519}{607.6} \][/tex]
[tex]\[ h_Y \approx 2.5 \, \text{m} \][/tex]
4. Object Z:
[tex]\[ m_Z = 24 \, \text{kg}, \, \text{PE}_Z = 1,176 \, \text{J} \][/tex]
[tex]\[ h_Z = \frac{1,176}{24 \times 9.8} \][/tex]
[tex]\[ h_Z = \frac{1,176}{235.2} \][/tex]
[tex]\[ h_Z \approx 5.0 \, \text{m} \][/tex]
Now, comparing the heights calculated:
- [tex]\( h_W \approx 2.0 \, \text{m} \)[/tex]
- [tex]\( h_X \approx 3.0 \, \text{m} \)[/tex]
- [tex]\( h_Y \approx 2.5 \, \text{m} \)[/tex]
- [tex]\( h_Z \approx 5.0 \, \text{m} \)[/tex]
The tallest height among these is [tex]\( 5.0 \, \text{m} \)[/tex].
Therefore, Object Z is on the tallest hill.
