Bài 1.
a)Bảo toàn cơ năng: \(W_1=W_2\)
\(\Rightarrow mgh+\dfrac{1}{2}mv_1^2=\dfrac{1}{2}mv_2^2\Rightarrow gh+\dfrac{1}{2}v_1^2=\dfrac{1}{2}v_2^2\)
\(\Rightarrow10\cdot h+\dfrac{1}{2}\cdot20^2=\dfrac{1}{2}\cdot30^2\Rightarrow h=25m\)
b)Độ cao vật cực đại so với mặt đất: \(W_2=W_3\)
\(\Rightarrow\dfrac{1}{2}mv^2_2=mgh_{max}\Rightarrow\dfrac{1}{2}v_2^2=g\cdot h\)
\(\Rightarrow\dfrac{1}{2}\cdot30^2=10\cdot h_{max}\Rightarrow h_{max}=45m\)
c)Động năng bằng 3 lần thế năng: \(W_đ=3W_t\Rightarrow W_đ=\dfrac{3}{4}W_t\)
\(\Rightarrow\dfrac{1}{2}mv'^2=\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot mv_2^2\Rightarrow v'^2=\dfrac{3}{4}\cdot v_2^2=\dfrac{3}{4}\cdot30^2=675\)
\(\Rightarrow v'=15\sqrt{3}\left(m/s\right)\)
Bài 2.
a)Độ cao cực đại so với mặt đất: \(\dfrac{1}{2}mv^2+mgh=mgh_{max}\)
\(\Rightarrow\dfrac{1}{2}v^2+gh=gh_{max}\Rightarrow\dfrac{1}{2}\cdot10^2+10\cdot10=h_{max}\Rightarrow h_{max}=15m\)
b)\(W_đ=3W_t\Rightarrow W_t=\dfrac{1}{4}W\)
\(\Rightarrow mgh'=\dfrac{1}{4}\cdot mgh_{max}\Rightarrow10\cdot h'=\dfrac{1}{4}\cdot10\cdot15\Rightarrow h'=3,75m\)
c)\(W_đ=W_t\Rightarrow W_đ=\dfrac{1}{2}W\Rightarrow\dfrac{1}{2}mv'^2=\dfrac{1}{2}\cdot mgh_{max}\)
\(\Rightarrow v'^2=10\cdot15=150\Rightarrow v'=5\sqrt{6}m/s\)
d)Vận tốc vật trước khi chạm đất:
\(v'=2\sqrt{gh_{max}}=2\sqrt{10\cdot15}=10\sqrt{6}m/s\)
Bài 3.
a)Động năng: \(W_đ=\dfrac{1}{2}mv^2=\dfrac{1}{2}\cdot0,02\cdot4^2=0,08J\)
Thế năng: \(W_t=mgz=0,02\cdot10\cdot1,6=0,32J\)
Cơ năng: \(W=W_đ+W_t=0,08+0,32=0,4J\)
b)Bảo toàn cơ năng: \(W=W_1\)
\(\Rightarrow0,4=mgh_{max}\Rightarrow h_{max}=2m\)
c)\(W_t=W_đ\Rightarrow W_t=\dfrac{1}{2}W\Rightarrow mgh'=\dfrac{1}{2}\cdot0,4\)
\(\Rightarrow h'=1m\)
