Question
While learning about energy transfers in science class, students worked in groups to design four different rockets. The goal of each rocket was to achieve as much potential energy as possible while spending as little money as possible on the materials. The characteristics of each rocket is listed in the table below.
Mass Maximum Height
Reached Total Cost
Rocket W 2 kg 22 m $17
Rocket X 2.5 kg 20 m $20
Rocket Y 1.5 kg 25 m $15
Rocket Z 5 kg 10 m $25
Based on the information given, which rocket best satisfied the criterion and constraint of the project?
A.
rocket W
B.
rocket Z
C.
rocket Y
D.
rocket X
Mass Maximum Height
Reached Total Cost
Rocket W 2 kg 22 m $17
Rocket X 2.5 kg 20 m $20
Rocket Y 1.5 kg 25 m $15
Rocket Z 5 kg 10 m $25
Based on the information given, which rocket best satisfied the criterion and constraint of the project?
A.
rocket W
B.
rocket Z
C.
rocket Y
D.
rocket X
Answers
GPT-4o mini
To determine which rocket best satisfied the criterion of achieving the maximum potential energy while spending as little money as possible, we need to calculate the potential energy (PE) for each rocket. The formula for potential energy is:
\[ \text{Potential Energy (PE)} = \text{mass} \times \text{height} \times g \]
where \(g\) (acceleration due to gravity) is approximately \(9.81 \, \text{m/s}^2\). However, since we are comparing the rockets, we can use the following simplified formula for potential energy (ignoring \(g\) since it is constant for all rockets):
\[ \text{PE} \propto \text{mass} \times \text{height} \]
Now we can compute this for each rocket:
1. **Rocket W:**
- Mass = 2 kg
- Maximum Height = 22 m
- PE = 2 kg × 22 m = **44**
2. **Rocket X:**
- Mass = 2.5 kg
- Maximum Height = 20 m
- PE = 2.5 kg × 20 m = **50**
3. **Rocket Y:**
- Mass = 1.5 kg
- Maximum Height = 25 m
- PE = 1.5 kg × 25 m = **37.5**
4. **Rocket Z:**
- Mass = 5 kg
- Maximum Height = 10 m
- PE = 5 kg × 10 m = **50**
Now we can compare the potential energy per dollar spent to determine which rocket is the best in terms of achieving maximum potential energy for the cost.
- **Rocket W:**
- PE = 44
- Cost = $17
- PE per dollar = 44 / 17 ≈ **2.59**
- **Rocket X:**
- PE = 50
- Cost = $20
- PE per dollar = 50 / 20 = **2.50**
- **Rocket Y:**
- PE = 37.5
- Cost = $15
- PE per dollar = 37.5 / 15 = **2.50**
- **Rocket Z:**
- PE = 50
- Cost = $25
- PE per dollar = 50 / 25 = **2.00**
From the calculations, Rocket W has the highest potential energy per dollar spent, followed closely by Rocket X and Rocket Y. Although Rocket X and Rocket Z have higher potential energy, their cost is relatively higher, which lowers their efficiency.
Based on this analysis, **Rocket W** best satisfies the criterion of achieving as much potential energy as possible while spending as little money as possible.
**Answer: A. rocket W**
\[ \text{Potential Energy (PE)} = \text{mass} \times \text{height} \times g \]
where \(g\) (acceleration due to gravity) is approximately \(9.81 \, \text{m/s}^2\). However, since we are comparing the rockets, we can use the following simplified formula for potential energy (ignoring \(g\) since it is constant for all rockets):
\[ \text{PE} \propto \text{mass} \times \text{height} \]
Now we can compute this for each rocket:
1. **Rocket W:**
- Mass = 2 kg
- Maximum Height = 22 m
- PE = 2 kg × 22 m = **44**
2. **Rocket X:**
- Mass = 2.5 kg
- Maximum Height = 20 m
- PE = 2.5 kg × 20 m = **50**
3. **Rocket Y:**
- Mass = 1.5 kg
- Maximum Height = 25 m
- PE = 1.5 kg × 25 m = **37.5**
4. **Rocket Z:**
- Mass = 5 kg
- Maximum Height = 10 m
- PE = 5 kg × 10 m = **50**
Now we can compare the potential energy per dollar spent to determine which rocket is the best in terms of achieving maximum potential energy for the cost.
- **Rocket W:**
- PE = 44
- Cost = $17
- PE per dollar = 44 / 17 ≈ **2.59**
- **Rocket X:**
- PE = 50
- Cost = $20
- PE per dollar = 50 / 20 = **2.50**
- **Rocket Y:**
- PE = 37.5
- Cost = $15
- PE per dollar = 37.5 / 15 = **2.50**
- **Rocket Z:**
- PE = 50
- Cost = $25
- PE per dollar = 50 / 25 = **2.00**
From the calculations, Rocket W has the highest potential energy per dollar spent, followed closely by Rocket X and Rocket Y. Although Rocket X and Rocket Z have higher potential energy, their cost is relatively higher, which lowers their efficiency.
Based on this analysis, **Rocket W** best satisfies the criterion of achieving as much potential energy as possible while spending as little money as possible.
**Answer: A. rocket W**