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Giải bài 9.16 trang 60 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Chuyển động của một hạt trên một dây rung được cho bởi công thức \(s\left( t \right) = 10 + \sqrt 2 \sin \left( {4\pi t + \frac{\pi }{6}} \right)\)

Tổng hợp đề thi họღc kì 2 lớp 11 tất cả các 🐎môn - Kết nối tri thức

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Chuyển động của một hạt trên một dây rung được cho bởi công thức \(s\left( t \right) = 10 + \sqrt 2 \sin \left( {4\pi t + \frac{\pi }{6}} \right)\), trong đó \(s\) tính bằng centimét và \(t\) tính bằng giây. Tính vận tốc của hạt sau \(t\) giây. Vận tốc cực đại của hạt là bao nhiêu? (Làm tròn kết quả đến chữ số thập phân thứ nhất).

Phương pháp giải - Xem chi tiết

Áp dụng công thức \(v\left( t \right) = s'(t)\) Vận tốc của hạt sau \(t\) giây là: \(v\left( t \right) = s'\left( t \right) = 4\pi \sqrt 2 \cos \left( {4\pi t + \frac{\pi }{6}} \right)\). Áp dụng tính chất \(\left| {\cos \left( {4\pi t + \frac{\pi }{6}} \right)} \right| \le 1\) Vận tốc cực đại của hạt là: \({v_{\max }} = 4\pi \sqrt 2  \approx 17,8\,\;{\rm{m}}/{\rm{s}}\), đạt được khi: \(\left| {\cos \left( {4\pi t + \frac{\pi }{6}} \right)} \right| = 1\)

Lời giải chi tiết

Vận tốc của hạt sau \(t\) giây là: \(v\left( t \right) = s'\left( t \right) = 4\pi \sqrt 2 {\rm{cos}}\left( {4\pi t + \frac{\pi }{6}} \right)\).Vận tốc cực đại của hạt là: \({v_{{\rm{max}}}} = 4\pi \sqrt 2  \approx 17,8{\rm{\;m}}/{\rm{s}}\), đạt được khi:\(\left| {{\rm{cos}}\left( {4\pi t + \frac{\pi }{6}} \right)} \right| = 1 \Leftrightarrow 4\pi t + \frac{\pi }{6} = \pi  + k\pi  \Leftrightarrow t = \frac{5}{{24}} + \frac{k}{4},k \in \mathbb{N}.\)

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