To find the new speed of the aircraft with respect to the ground, we need to perform vector addition of the aircraft's velocity and the wind's velocity.
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Set up the vectors:
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The aircraft's velocity vector (\( \vec{V}{\text{plane}} \)) can be represented as: \[ \vec{V}{\text{plane}} = (915, 0) \] (since it's moving east, there's no northward component).
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The wind's velocity vector (\( \vec{V}{\text{wind}} \)) at a speed of 969 mph, 34 degrees north of east can be broken down into its components: \[ \vec{V}{\text{wind}} = (969 \cos(34^\circ), 969 \sin(34^\circ)) \]
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Calculate the components of the wind’s velocity:
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We first calculate \( \cos(34^\circ) \) and \( \sin(34^\circ) \):
- Using a calculator, we find: \[ \cos(34^\circ) \approx 0.8290 \] \[ \sin(34^\circ) \approx 0.5592 \]
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Now, calculate the components of the wind: \[ V_{\text{wind}, x} = 969 \cdot \cos(34^\circ) \approx 969 \cdot 0.8290 \approx 805.661 \] \[ V_{\text{wind}, y} = 969 \cdot \sin(34^\circ) \approx 969 \cdot 0.5592 \approx 542.229 \]
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Therefore, the wind velocity vector is approximately: \[ \vec{V}_{\text{wind}} \approx (805.661, 542.229) \]
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Add the velocity vectors:
- Now we add the two velocity vectors: \[ \vec{V}{\text{total}} = \vec{V}{\text{plane}} + \vec{V}_{\text{wind}} = (915, 0) + (805.661, 542.229) = (1720.661, 542.229) \]
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Calculate the magnitude of the resultant velocity:
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The magnitude of the resultant vector is given by: \[ V_{\text{total}} = \sqrt{(1720.661)^2 + (542.229)^2} \]
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Calculating each component: \[ (1720.661)^2 \approx 2951586.578 \] \[ (542.229)^2 \approx 294805.087 \]
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Now sum these values: \[ V_{\text{total}} = \sqrt{2951586.578 + 294805.087} \approx \sqrt{3246391.665} \approx 1803.55 \]
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Final answer:
- Therefore, the new speed of the aircraft with respect to the ground is approximately: \[ \boxed{1803.55} \text{ mph} \]
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