Answer:
So, the magnitude of the velocity of the plane is approximately 191.7 m/s, and the direction is approximately \(83.7^\circ\) above the positive x-axis.
Explanation:
1. The plane's velocity is given as 190 m/s in the y-direction. This means the plane is moving vertically upward or downward, directly up or down.
2. The wind is blowing at 25.0 m/s in the x-direction. This means the wind is pushing horizontally, to the left or right.
To find the overall velocity of the plane, we need to combine these two velocities, taking into account both their magnitude (speed) and direction. We can use vector addition for this.
Imagine the plane's velocity as one arrow pointing straight up or down, and the wind's velocity as another arrow pointing horizontally. The resulting velocity of the plane is the diagonal of the parallelogram formed by these two arrows.
Now, we can use the Pythagorean theorem to find the magnitude (speed) of the resulting velocity:
\[ \text{Magnitude} = \sqrt{(190 \, \text{m/s})^2 + (25 \, \text{m/s})^2} \]
\[ \text{Magnitude} = \sqrt{36100 + 625} \]
\[ \text{Magnitude} = \sqrt{36725} \]
\[ \text{Magnitude} ≈ 191.7 \, \text{m/s} \]
To find the direction, we can use trigonometry. The direction is the angle that the resulting velocity makes with the positive x-axis. We can use the inverse tangent function:
\[ \text{Direction} = \tan^{-1}\left(\frac{\text{vertical component}}{\text{horizontal component}}\right) \]
\[ \text{Direction} = \tan^{-1}\left(\frac{190}{25}\right) \]
\[ \text{Direction} = \tan^{-1}(7.6) \]
\[ \text{Direction} ≈ 83.7^\circ \]
So, the magnitude of the velocity of the plane is approximately 191.7 m/s, and the direction is approximately \(83.7^\circ\) above the positive x-axis.
Answer:
191.6 m/s at 82.5° above the +x axis
Explanation:
The magnitude of a vector can be found using the x and y components of the vector and Pythagorean theorem. The direction of the vector can be found using trigonometry.
The y component of the plane's velocity is 190 m/s. The x component of the plane's velocity is 25.0 m/s. Using Pythagorean theorem, the plane's total velocity is:
v² = vₓ² + vᵧ²
v² = (25.0)² + (190)²
v = 191.6 m/s
The direction of the plane's velocity can be found using trigonometry.
tan θ = vᵧ / vₓ
tan θ = 190 / 25.0
θ = 82.5°
The plane's velocity is 191.6 m/s at 82.5° above the +x axis.
