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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.
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