Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine which combination of the rocket body and engine will result in an acceleration of [tex]\( 40 \, \text{m/s}^2 \)[/tex] at the start of the launch, we can use Newton's second law of motion. Newton's second law states:
[tex]\[ F = ma \][/tex]
where:
- [tex]\( F \)[/tex] is the force applied (in Newtons, N),
- [tex]\( m \)[/tex] is the mass of the object (in kilograms, kg),
- [tex]\( a \)[/tex] is the acceleration (in meters per second squared, [tex]\(\text{m/s}^2\)[/tex]).
We are given the following:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Body} & \text{Mass (kg)} & \text{Engine} & \text{Force (N)} \\ \hline 1 & 0.50 & 1 & 25 \\ \hline 2 & 1.5 & 2 & 20 \\ \hline 3 & 0.75 & 3 & 30 \\ \hline \end{array} \][/tex]
To find the acceleration, we rearrange the formula to solve for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{F}{m} \][/tex]
We need to check each combination of rocket body and engine to see which one results in an acceleration of [tex]\( 40 \, \text{m/s}^2 \)[/tex].
### Combination Calculations:
1. Body 1 + Engine 1:
[tex]\[ a = \frac{25 \, \text{N}}{0.50 \, \text{kg}} = 50 \, \text{m/s}^2 \][/tex]
2. Body 2 + Engine 2:
[tex]\[ a = \frac{20 \, \text{N}}{1.5 \, \text{kg}} = \frac{20}{1.5} \approx 13.33 \, \text{m/s}^2 \][/tex]
3. Body 3 + Engine 3:
[tex]\[ a = \frac{30 \, \text{N}}{0.75 \, \text{kg}} = 40 \, \text{m/s}^2 \][/tex]
4. Body 1 + Engine 2:
[tex]\[ a = \frac{20 \, \text{N}}{0.50 \, \text{kg}} = 40 \, \text{m/s}^2 \][/tex]
5. Body 1 + Engine 3:
[tex]\[ a = \frac{30 \, \text{N}}{0.50 \, \text{kg}} = 60 \, \text{m/s}^2 \][/tex]
6. Body 2 + Engine 1:
[tex]\[ a = \frac{25 \, \text{N}}{1.5 \, \text{kg}} \approx 16.67 \, \text{m/s}^2 \][/tex]
7. Body 3 + Engine 2:
[tex]\[ a = \frac{20 \, \text{N}}{0.75 \, \text{kg}} = \frac{20}{0.75} \approx 26.67 \, \text{m/s}^2 \][/tex]
8. Body 2 + Engine 3:
[tex]\[ a = \frac{30 \, \text{N}}{1.5 \, \text{kg}} = 20 \, \text{m/s}^2 \][/tex]
9. Body 3 + Engine 1:
[tex]\[ a = \frac{25 \, \text{N}}{0.75 \, \text{kg}} \approx 33.33 \, \text{m/s}^2 \][/tex]
From our calculations, the combinations that result in an acceleration of [tex]\( 40 \, \text{m/s}^2 \)[/tex] are:
- Body 3 + Engine 3
- Body 1 + Engine 2
Therefore, the best combination with [tex]\( 40 \, \text{m/s}^2 \)[/tex] acceleration considering practical solutions would be the combination Body 1 + Engine 2.
[tex]\[ F = ma \][/tex]
where:
- [tex]\( F \)[/tex] is the force applied (in Newtons, N),
- [tex]\( m \)[/tex] is the mass of the object (in kilograms, kg),
- [tex]\( a \)[/tex] is the acceleration (in meters per second squared, [tex]\(\text{m/s}^2\)[/tex]).
We are given the following:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Body} & \text{Mass (kg)} & \text{Engine} & \text{Force (N)} \\ \hline 1 & 0.50 & 1 & 25 \\ \hline 2 & 1.5 & 2 & 20 \\ \hline 3 & 0.75 & 3 & 30 \\ \hline \end{array} \][/tex]
To find the acceleration, we rearrange the formula to solve for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{F}{m} \][/tex]
We need to check each combination of rocket body and engine to see which one results in an acceleration of [tex]\( 40 \, \text{m/s}^2 \)[/tex].
### Combination Calculations:
1. Body 1 + Engine 1:
[tex]\[ a = \frac{25 \, \text{N}}{0.50 \, \text{kg}} = 50 \, \text{m/s}^2 \][/tex]
2. Body 2 + Engine 2:
[tex]\[ a = \frac{20 \, \text{N}}{1.5 \, \text{kg}} = \frac{20}{1.5} \approx 13.33 \, \text{m/s}^2 \][/tex]
3. Body 3 + Engine 3:
[tex]\[ a = \frac{30 \, \text{N}}{0.75 \, \text{kg}} = 40 \, \text{m/s}^2 \][/tex]
4. Body 1 + Engine 2:
[tex]\[ a = \frac{20 \, \text{N}}{0.50 \, \text{kg}} = 40 \, \text{m/s}^2 \][/tex]
5. Body 1 + Engine 3:
[tex]\[ a = \frac{30 \, \text{N}}{0.50 \, \text{kg}} = 60 \, \text{m/s}^2 \][/tex]
6. Body 2 + Engine 1:
[tex]\[ a = \frac{25 \, \text{N}}{1.5 \, \text{kg}} \approx 16.67 \, \text{m/s}^2 \][/tex]
7. Body 3 + Engine 2:
[tex]\[ a = \frac{20 \, \text{N}}{0.75 \, \text{kg}} = \frac{20}{0.75} \approx 26.67 \, \text{m/s}^2 \][/tex]
8. Body 2 + Engine 3:
[tex]\[ a = \frac{30 \, \text{N}}{1.5 \, \text{kg}} = 20 \, \text{m/s}^2 \][/tex]
9. Body 3 + Engine 1:
[tex]\[ a = \frac{25 \, \text{N}}{0.75 \, \text{kg}} \approx 33.33 \, \text{m/s}^2 \][/tex]
From our calculations, the combinations that result in an acceleration of [tex]\( 40 \, \text{m/s}^2 \)[/tex] are:
- Body 3 + Engine 3
- Body 1 + Engine 2
Therefore, the best combination with [tex]\( 40 \, \text{m/s}^2 \)[/tex] acceleration considering practical solutions would be the combination Body 1 + Engine 2.
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.