Final answer:
To find the magnitude of the resultant force exerted by the two engines, we can use vector addition. The forward thrust of 675 N does not have any vertical component, so we can ignore it when calculating the vertical component of the resultant force. The thrust of 509 N at an angle of 30.4 degrees above the forward direction has a vertical component of 262.235 N and a horizontal component of 1119.266 N. Using the Pythagorean theorem, the magnitude of the resultant force is approximately 1161 N.
Step-by-step explanation:
To find the magnitude of the resultant force exerted by the two engines, we can use vector addition. We can break down the forces into their horizontal and vertical components. The forward thrust of 675 N does not have any vertical component, so we can ignore it when calculating the vertical component of the resultant force. The thrust of 509 N at an angle of 30.4 degrees above the forward direction has a vertical component of 509 N * sin(30.4) and a horizontal component of 509 N * cos(30.4). Now, we can add the vertical components of the two forces to find the total vertical force:
509 N * sin(30.4) + 0 = 509 N * 0.515 + 0 = 262.235 N
Then, we can add the horizontal components of the two forces to find the total horizontal force:
675 N + 509 N * cos(30.4) = 675 N + 509 N * 0.874 = 675 N + 444.266 N = 1119.266 N
Finally, we can use the Pythagorean theorem to find the magnitude of the resultant force:
Magnitude of resultant force = sqrt((1119.266 N)^2 + (262.235 N)^2) = sqrt(1347928.303 N^2) ≈ 1161 N
Therefore, the magnitude of the resultant force exerted by the engines on the rocket is approximately 1161 N.