Answer :
Let's solve this problem step-by-step to determine which object is on the tallest hill.
Step 1: Understand the Potential Energy formula
The potential energy ([tex]\( PE \)[/tex]) of an object is given by the formula:
[tex]\[ PE = m \cdot g \cdot h \][/tex]
where,
- [tex]\( PE \)[/tex] is the potential energy,
- [tex]\( m \)[/tex] is the mass of the object,
- [tex]\( g \)[/tex] is the gravitational acceleration (approximated as [tex]\( 9.8 \, \text{m/s}^2 \)[/tex]),
- [tex]\( h \)[/tex] is the height of the hill.
Step 2: Rearrange the formula to solve for height ([tex]\( h \)[/tex])
To find the height, we rearrange the formula:
[tex]\[ h = \frac{PE}{m \cdot g} \][/tex]
Step 3: Calculate the height for each object
Given:
1. Object W:
- Mass ([tex]\( m \)[/tex]) = 50 kg
- Potential energy ([tex]\( PE \)[/tex]) = 980 J
[tex]\[ h_W = \frac{980}{50 \cdot 9.8} = \frac{980}{490} \approx 2 \, \text{m} \][/tex]
2. Object X:
- Mass ([tex]\( m \)[/tex]) = 35 kg
- Potential energy ([tex]\( PE \)[/tex]) = 1,029 J
[tex]\[ h_X = \frac{1,029}{35 \cdot 9.8} = \frac{1,029}{343} \approx 3 \, \text{m} \][/tex]
3. Object Y:
- Mass ([tex]\( m \)[/tex]) = 62 kg
- Potential energy ([tex]\( PE \)[/tex]) = 1,519 J
[tex]\[ h_Y = \frac{1,519}{62 \cdot 9.8} = \frac{1,519}{607.6} \approx 2.5 \, \text{m} \][/tex]
4. Object Z:
- Mass ([tex]\( m \)[/tex]) = 24 kg
- Potential energy ([tex]\( PE \)[/tex]) = 1,176 J
[tex]\[ h_Z = \frac{1,176}{24 \cdot 9.8} = \frac{1,176}{235.2} \approx 5 \, \text{m} \][/tex]
Step 4: Compare the heights to determine which hill is tallest
The heights for each object are:
- [tex]\( h_W \approx 2 \, \text{m} \)[/tex]
- [tex]\( h_X \approx 3 \, \text{m} \)[/tex]
- [tex]\( h_Y \approx 2.5 \, \text{m} \)[/tex]
- [tex]\( h_Z \approx 5 \, \text{m} \)[/tex]
Comparing these heights, we find that [tex]\( h_Z \approx 5 \text{m} \)[/tex] is the tallest.
Conclusion:
The object on the tallest hill is object [tex]\( Z \)[/tex].
So, the answer is:
[tex]\[ \boxed{Z} \][/tex]
Step 1: Understand the Potential Energy formula
The potential energy ([tex]\( PE \)[/tex]) of an object is given by the formula:
[tex]\[ PE = m \cdot g \cdot h \][/tex]
where,
- [tex]\( PE \)[/tex] is the potential energy,
- [tex]\( m \)[/tex] is the mass of the object,
- [tex]\( g \)[/tex] is the gravitational acceleration (approximated as [tex]\( 9.8 \, \text{m/s}^2 \)[/tex]),
- [tex]\( h \)[/tex] is the height of the hill.
Step 2: Rearrange the formula to solve for height ([tex]\( h \)[/tex])
To find the height, we rearrange the formula:
[tex]\[ h = \frac{PE}{m \cdot g} \][/tex]
Step 3: Calculate the height for each object
Given:
1. Object W:
- Mass ([tex]\( m \)[/tex]) = 50 kg
- Potential energy ([tex]\( PE \)[/tex]) = 980 J
[tex]\[ h_W = \frac{980}{50 \cdot 9.8} = \frac{980}{490} \approx 2 \, \text{m} \][/tex]
2. Object X:
- Mass ([tex]\( m \)[/tex]) = 35 kg
- Potential energy ([tex]\( PE \)[/tex]) = 1,029 J
[tex]\[ h_X = \frac{1,029}{35 \cdot 9.8} = \frac{1,029}{343} \approx 3 \, \text{m} \][/tex]
3. Object Y:
- Mass ([tex]\( m \)[/tex]) = 62 kg
- Potential energy ([tex]\( PE \)[/tex]) = 1,519 J
[tex]\[ h_Y = \frac{1,519}{62 \cdot 9.8} = \frac{1,519}{607.6} \approx 2.5 \, \text{m} \][/tex]
4. Object Z:
- Mass ([tex]\( m \)[/tex]) = 24 kg
- Potential energy ([tex]\( PE \)[/tex]) = 1,176 J
[tex]\[ h_Z = \frac{1,176}{24 \cdot 9.8} = \frac{1,176}{235.2} \approx 5 \, \text{m} \][/tex]
Step 4: Compare the heights to determine which hill is tallest
The heights for each object are:
- [tex]\( h_W \approx 2 \, \text{m} \)[/tex]
- [tex]\( h_X \approx 3 \, \text{m} \)[/tex]
- [tex]\( h_Y \approx 2.5 \, \text{m} \)[/tex]
- [tex]\( h_Z \approx 5 \, \text{m} \)[/tex]
Comparing these heights, we find that [tex]\( h_Z \approx 5 \text{m} \)[/tex] is the tallest.
Conclusion:
The object on the tallest hill is object [tex]\( Z \)[/tex].
So, the answer is:
[tex]\[ \boxed{Z} \][/tex]
