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The Rocket Club is planning to launch a pair of model rockets. To build the rocket, the club needs a rocket body paired with an engine. The table lists the mass of three possible rocket bodies and the force generated by three possible engines.

| Body | Mass (kg) | Engine | Force (N) |
|------|-----------|--------|-----------|
| 1 | 0.500 | 1 | 25 |
| 2 | 1.5 | 2 | 20 |
| 3 | 0.750 | 3 | 30 |

Based on Newton's laws of motion, which combination of rocket bodies and engines will result in the acceleration of 40 m/s² at the start of the launch?

A. Body 3 + Engine 1
B. Body 2 + Engine 2
C. Body 1 + Engine 2
D. Body 1 + Engine 1


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.
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