To find which combinations of rocket bodies and engines will result in an acceleration of [tex]\( 40 \, m/s² \)[/tex], we must use Newton's second law of motion which states [tex]\( F = ma \)[/tex]. Rearranging this equation to solve for acceleration [tex]\( a \)[/tex], we get:
[tex]\[ a = \frac{F}{m} \][/tex]
We will calculate the acceleration for each combination of rocket bodies and engines and find which one results in an acceleration of [tex]\( 40 \, m/s² \)[/tex].
1. Body 1 + Engine 1
- Mass ([tex]\( m \)[/tex]) = 0.500 kg
- Force ([tex]\( F \)[/tex]) = 25 N
[tex]\[ a = \frac{25 \, \text{N}}{0.500 \, \text{kg}} = 50 \, m/s² \][/tex]
2. Body 2 + Engine 2
- Mass ([tex]\( m \)[/tex]) = 1.5 kg
- Force ([tex]\( F \)[/tex]) = 20 N
[tex]\[ a = \frac{20 \, \text{N}}{1.5 \, \text{kg}} = 13.33 \, m/s² \][/tex]
3. Body 3 + Engine 3
- Mass ([tex]\( m \)[/tex]) = 0.750 kg
- Force ([tex]\( F \)[/tex]) = 30 N
[tex]\[ a = \frac{30 \, \text{N}}{0.750 \, \text{kg}} = 40 \, m/s² \][/tex]
4. Body 1 + Engine 2
- Mass ([tex]\( m \)[/tex]) = 0.500 kg
- Force ([tex]\( F \)[/tex]) = 20 N
[tex]\[ a = \frac{20 \, \text{N}}{0.500 \, \text{kg}} = 40 \, m/s² \][/tex]
5. Body 2 + Engine 1
- Mass ([tex]\( m \)[/tex]) = 1.5 kg
- Force ([tex]\( F \)[/tex]) = 25 N
[tex]\[ a = \frac{25 \, \text{N}}{1.5 \, \text{kg}} = 16.67 \, m/s² \][/tex]
6. Body 3 + Engine 1
- Mass ([tex]\( m \)[/tex]) = 0.750 kg
- Force ([tex]\( F \)[/tex]) = 25 N
[tex]\[ a = \frac{25 \, \text{N}}{0.750 \, \text{kg}} = 33.33 \, m/s² \][/tex]
7. Body 1 + Engine 3
- Mass ([tex]\( m \)[/tex]) = 0.500 kg
- Force ([tex]\( F \)[/tex]) = 30 N
[tex]\[ a = \frac{30 \, \text{N}}{0.500 \, \text{kg}} = 60 \, m/s² \][/tex]
8. Body 2 + Engine 3
- Mass ([tex]\( m \)[/tex]) = 1.5 kg
- Force ([tex]\( F \)[/tex]) = 30 N
[tex]\[ a = \frac{30 \, \text{N}}{1.5 \, \text{kg}} = 20 \, m/s² \][/tex]
9. Body 3 + Engine 2
- Mass ([tex]\( m \)[/tex]) = 0.750 kg
- Force ([tex]\( F \)[/tex]) = 20 N
[tex]\[ a = \frac{20 \, \text{N}}{0.750 \, \text{kg}} = 26.67 \, m/s² \][/tex]
From these calculations, we find that the combinations resulting in an acceleration of [tex]\( 40 \, m/s² \)[/tex] are:
- Body 1 + Engine 2
- Body 3 + Engine 3
Therefore, the combinations that will achieve the required acceleration are [tex]\( \boxed{\text{Body 1 + Engine 2 and Body 3 + Engine 3}} \)[/tex].
