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 Z
B.
rocket X
C.
rocket W
D.
rocket Y
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 Z
B.
rocket X
C.
rocket W
D.
rocket Y
Answers
GPT-4o mini
To determine which rocket best satisfies the criteria of achieving maximum potential energy while minimizing cost, we can calculate the potential energy for each rocket and compare it in relation to its cost.
The potential energy (PE) can be calculated using the formula:
\[ PE = mgh \]
where:
- \( m \) = mass (in kg)
- \( g \) = acceleration due to gravity (approximately \( 9.8 \, m/s^2 \))
- \( h \) = maximum height reached (in m)
Let's calculate the potential energy for each rocket:
1. **Rocket W**
- Mass \( m = 2 \, kg \)
- Height \( h = 22 \, m \)
- PE = \( 2 \, kg \times 9.8 \, m/s^2 \times 22 \, m = 431.2 \, J \)
- Cost = $17
- PE per dollar = \( \frac{431.2 \, J}{17} \approx 25.4 \, J/\$ \)
2. **Rocket X**
- Mass \( m = 2.5 \, kg \)
- Height \( h = 20 \, m \)
- PE = \( 2.5 \, kg \times 9.8 \, m/s^2 \times 20 \, m = 490 \, J \)
- Cost = $20
- PE per dollar = \( \frac{490 \, J}{20} = 24.5 \, J/\$ \)
3. **Rocket Y**
- Mass \( m = 1.5 \, kg \)
- Height \( h = 25 \, m \)
- PE = \( 1.5 \, kg \times 9.8 \, m/s^2 \times 25 \, m = 367.5 \, J \)
- Cost = $15
- PE per dollar = \( \frac{367.5 \, J}{15} = 24.5 \, J/\$ \)
4. **Rocket Z**
- Mass \( m = 5 \, kg \)
- Height \( h = 10 \, m \)
- PE = \( 5 \, kg \times 9.8 \, m/s^2 \times 10 \, m = 490 \, J \)
- Cost = $25
- PE per dollar = \( \frac{490 \, J}{25} = 19.6 \, J/\$ \)
Now let's summarize the PE per dollar:
- Rocket W: \( \approx 25.4 \, J/\$ \)
- Rocket X: \( 24.5 \, J/\$ \)
- Rocket Y: \( 24.5 \, J/\$ \)
- Rocket Z: \( 19.6 \, J/\$ \)
Rocket W has the highest potential energy per dollar spent.
**Therefore, the rocket that best satisfied the criterion of achieving maximum potential energy for the least cost is:**
**C. Rocket W**
The potential energy (PE) can be calculated using the formula:
\[ PE = mgh \]
where:
- \( m \) = mass (in kg)
- \( g \) = acceleration due to gravity (approximately \( 9.8 \, m/s^2 \))
- \( h \) = maximum height reached (in m)
Let's calculate the potential energy for each rocket:
1. **Rocket W**
- Mass \( m = 2 \, kg \)
- Height \( h = 22 \, m \)
- PE = \( 2 \, kg \times 9.8 \, m/s^2 \times 22 \, m = 431.2 \, J \)
- Cost = $17
- PE per dollar = \( \frac{431.2 \, J}{17} \approx 25.4 \, J/\$ \)
2. **Rocket X**
- Mass \( m = 2.5 \, kg \)
- Height \( h = 20 \, m \)
- PE = \( 2.5 \, kg \times 9.8 \, m/s^2 \times 20 \, m = 490 \, J \)
- Cost = $20
- PE per dollar = \( \frac{490 \, J}{20} = 24.5 \, J/\$ \)
3. **Rocket Y**
- Mass \( m = 1.5 \, kg \)
- Height \( h = 25 \, m \)
- PE = \( 1.5 \, kg \times 9.8 \, m/s^2 \times 25 \, m = 367.5 \, J \)
- Cost = $15
- PE per dollar = \( \frac{367.5 \, J}{15} = 24.5 \, J/\$ \)
4. **Rocket Z**
- Mass \( m = 5 \, kg \)
- Height \( h = 10 \, m \)
- PE = \( 5 \, kg \times 9.8 \, m/s^2 \times 10 \, m = 490 \, J \)
- Cost = $25
- PE per dollar = \( \frac{490 \, J}{25} = 19.6 \, J/\$ \)
Now let's summarize the PE per dollar:
- Rocket W: \( \approx 25.4 \, J/\$ \)
- Rocket X: \( 24.5 \, J/\$ \)
- Rocket Y: \( 24.5 \, J/\$ \)
- Rocket Z: \( 19.6 \, J/\$ \)
Rocket W has the highest potential energy per dollar spent.
**Therefore, the rocket that best satisfied the criterion of achieving maximum potential energy for the least cost is:**
**C. Rocket W**