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Question 4: Passengers on the deck of a boat with a speed of v, = 20 m/s with the given direction experience a wind blowing at a speed of vw = 10 m/s as shown. What is the magnitude of the wind speed experienced by someone on the shore? V, = 20 m/s 30° 45° Vw = 10 m/s y 30 ee X

Question 4: Passengers on the deck of a boat with a speed of v, = 20 m/s with the given direction experience a wind blowing at a speed of vw = 10 m/s as shown. What is the magnitude of the wind speed experienced by someone on the shore? V, = 20 m/s 30° 45° Vw = 10 m/s y 30 ee X

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### Question 4 Passengers on the deck of a boat with a speed of \( v_s = 20 \, \text{m/s} \) with the given direction experience a wind blowing at a speed of \( v_w = 10 \, \text{m/s} \) as shown. What is the magnitude of the wind speed experienced by someone on the shore? ### Diagram Explanation The image shows a boat moving with a speed of \( v_s = 20 \, \text{m/s} \) at an angle of 45° relative to the x-axis. There is also a wind blowing with a speed of \( v_w = 10 \, \text{m/s} \) at an angle of 30° relative to the negative y-axis. #### Wind and Boat Velocity Components: - The boat's velocity (\( v_s \)) has components: - \( v_{sx} = v_s \cos(45^\circ) = 20 \cos(45^\circ) \) - \( v_{sy} = v_s \sin(45^\circ) = 20 \sin(45^\circ) \) - The wind's velocity (\( v_w \)) has components: - \( v_{wx} = v_w \cos(30^\circ) = 10 \cos(30^\circ) \) - \( v_{wy} = -v_w \sin(30^\circ) = -10 \sin(30^\circ) \) We need to determine the combined velocity of the wind from the perspective of someone on the shore, taking both the wind speed and the movement of the boat into account. This involves vector addition of the two velocities. #### Vector Addition of Velocities: - Net \( x \)-component: \( v_x = v_{sx} + v_{wx} \) - Net \( y \)-component: \( v_y = v_{sy} + v_{wy} \) Finally, calculate the magnitude of the resulting combined wind speed using the Pythagorean theorem: \[ v_{\text{net}} = \sqrt{(v_x)^2 + (v_y)^2} \] By following these steps, students can systematically determine the magnitude of the wind speed experienced by someone on the shore.
Transcribed Image Text:### Question 4 Passengers on the deck of a boat with a speed of \( v_s = 20 \, \text{m/s} \) with the given direction experience a wind blowing at a speed of \( v_w = 10 \, \text{m/s} \) as shown. What is the magnitude of the wind speed experienced by someone on the shore? ### Diagram Explanation The image shows a boat moving with a speed of \( v_s = 20 \, \text{m/s} \) at an angle of 45° relative to the x-axis. There is also a wind blowing with a speed of \( v_w = 10 \, \text{m/s} \) at an angle of 30° relative to the negative y-axis. #### Wind and Boat Velocity Components: - The boat's velocity (\( v_s \)) has components: - \( v_{sx} = v_s \cos(45^\circ) = 20 \cos(45^\circ) \) - \( v_{sy} = v_s \sin(45^\circ) = 20 \sin(45^\circ) \) - The wind's velocity (\( v_w \)) has components: - \( v_{wx} = v_w \cos(30^\circ) = 10 \cos(30^\circ) \) - \( v_{wy} = -v_w \sin(30^\circ) = -10 \sin(30^\circ) \) We need to determine the combined velocity of the wind from the perspective of someone on the shore, taking both the wind speed and the movement of the boat into account. This involves vector addition of the two velocities. #### Vector Addition of Velocities: - Net \( x \)-component: \( v_x = v_{sx} + v_{wx} \) - Net \( y \)-component: \( v_y = v_{sy} + v_{wy} \) Finally, calculate the magnitude of the resulting combined wind speed using the Pythagorean theorem: \[ v_{\text{net}} = \sqrt{(v_x)^2 + (v_y)^2} \] By following these steps, students can systematically determine the magnitude of the wind speed experienced by someone on the shore.
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