To determine which combination of rocket bodies and engines will result in an acceleration of [tex]\(40 \, \text{m/s}^2\)[/tex] at the start of the launch, we apply Newton's second law of motion. Newton's second law states that [tex]\( F = ma \)[/tex], where [tex]\( F \)[/tex] is the force applied to an object, [tex]\( m \)[/tex] is the mass of the object, and [tex]\( a \)[/tex] is its acceleration. This can be rearranged to find acceleration:
[tex]\[ a = \frac{F}{m} \][/tex]
We need to evaluate the possible combinations of rocket bodies and engines to determine which one provides the required acceleration of [tex]\(40 \, \text{m/s}^2\)[/tex].
Body 1 + Engine 1:
- Mass of Body 1: [tex]\(0.5 \, \text{kg}\)[/tex]
- Force from Engine 1: [tex]\(25 \, \text{N}\)[/tex]
- Acceleration: [tex]\( a = \frac{25 \, \text{N}}{0.5 \, \text{kg}} = 50 \, \text{m/s}^2 \)[/tex]
Body 2 + Engine 2:
- Mass of Body 2: [tex]\(1.5 \, \text{kg}\)[/tex]
- Force from Engine 2: [tex]\(20 \, \text{N}\)[/tex]
- Acceleration: [tex]\( a = \frac{20 \, \text{N}}{1.5 \, \text{kg}} = 13.33 \, \text{m/s}^2 \)[/tex] (rounded to two decimal places)
Body 1 + Engine 2:
- Mass of Body 1: [tex]\(0.5 \, \text{kg}\)[/tex]
- Force from Engine 2: [tex]\(20 \, \text{N}\)[/tex]
- Acceleration: [tex]\( a = \frac{20 \, \text{N}}{0.5 \, \text{kg}} = 40 \, \text{m/s}^2 \)[/tex]
Body 3 + Engine 1:
- Mass of Body 3: [tex]\(0.75 \, \text{kg}\)[/tex]
- Force from Engine 1: [tex]\(25 \, \text{N}\)[/tex]
- Acceleration: [tex]\( a = \frac{25 \, \text{N}}{0.75 \, \text{kg}} = 33.33 \, \text{m/s}^2 \)[/tex] (rounded to two decimal places)
By comparing the accelerations calculated for each combination, we see that:
- Body 1 + Engine 1 results in [tex]\(50 \, \text{m/s}^2\)[/tex]
- Body 2 + Engine 2 results in [tex]\(13.33 \, \text{m/s}^2\)[/tex] (rounded)
- Body 1 + Engine 2 results in [tex]\(40 \, \text{m/s}^2\)[/tex]
- Body 3 + Engine 1 results in [tex]\(33.33 \, \text{m/s}^2\)[/tex] (rounded)
The combination that results in the desired acceleration of [tex]\(40 \, \text{m/s}^2\)[/tex] is Body 1 + Engine 2.
Therefore, the correct answer is: Body 1 + Engine 2.
