Respuesta :
Answer:
Approximately [tex]6.3\; {\rm m\cdot s^{-2}}[/tex].
Explanation:
In this question, the following quantities form a right triangle:
- acceleration in the westward direction,
- acceleration in the southward direction, and
- the acceleration vector (pointing at [tex]33^{\circ}[/tex] south of west.)
The acceleration vector represents the hypotenuse of this right triangle. Additionally, it is given that acceleration points at [tex]33^{\circ}[/tex] south of west, meaning that the angle between this vector and the westward direction is [tex]\theta = 33^{\circ}[/tex].
The goal is to find acceleration in the westward direction given the magnitude of acceleration, [tex]7.5\; {\rm m\cdot s^{-2}}[/tex]. Note that acceleration in the westward direction is adjacent to the [tex]\theta = 33^{\circ}[/tex] angle in this right triangle, whereas the acceleration vector represents the hypotenuse. The ratio between the two sides can be found using trigonometry:
[tex]\begin{aligned} & \frac{\text{westward acceleration}}{\text{acceleration}} \\ =\; & \frac{\text{adjacent}}{\text{hypotenuse}} \\ =\; & \cos(\theta)\end{aligned}[/tex].
Rearrange to obtain:
[tex]\begin{aligned} & (\text{westward acceleration}) \\=\; & (\text{acceleration})\, \cos(\theta) \\ =\; & (7.5\; {\rm m\cdot s^{-2}})\, \cos(33^{\circ}) \\ \approx\; & 6.3\; {\rm m\cdot s^{-2}}\end{aligned}[/tex].
In other words, the acceleration in the westward direction would be approximately [tex]6.3\; {\rm m\cdot s^{-2}}[/tex].
Final answer:
The airplane's acceleration in the westward direction is calculated to be approximately 6.2 m/s² by resolving the total acceleration of 7.5 m/s² into components using the cosine of the angle 33° south of west.
Explanation:
To find the airplane's acceleration in the westward direction, we will use the component form of acceleration. The total acceleration is given as 7.5 m/s² at an angle of 33° south of west. To resolve this into components, we can use the cosine of the angle for the westward (horizontal) component, which will be:
Accelerationwest = Accelerationtotal × cos(Θ).
Let's do the calculations step by step:
Identify the total acceleration and the angle: Accelerationtotal = 7.5 m/s² and Θ = 33°.
Calculate the westward component using the cosine function:
Accelerationwest = 7.5 m/s² × cos(33°).
Using a calculator (assuming the cosine of 33° is approximately 0.838),
Accelerationwest = 7.5 m/s² × 0.838 ≈ 6.29 m/s², which rounds to 6.2 m/s² (closest to option b).
Therefore, the acceleration of the airplane in the westward direction is approximately 6.2 m/s².
