Движение тела, брошенного горизонтально.¶
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\[ \begin{align}\begin{aligned}\begin{split}\begin{cases} v_x = v_0 \\ v_y = -gt \end{cases}\end{split}\\\begin{split}\begin{cases} x = v_0 t\\ y = y_0 -\frac{g t^2}{2} \end{cases}\end{split}\\\begin{split}\begin{cases} t = \frac{x}{v_0} \\ y = y_0 - \frac{gx^2}{2v_0^2} \end{cases}\end{split}\\t = \frac{x}{v_0} \rightarrow y(x) = h_0 - \frac{gx^2}{2y^2}\\y = 0 \rightarrow h - \frac{gt^2}{2} = 0 \rightarrow t = \sqrt{\frac{2h_0}{g}}\\x_к = v_0 * \sqrt{\frac{2h_0}{g}}\\v_к = \sqrt{v_x ^ 2 + v_y ^ 2} = \sqrt{v_0 ^ 2 + (gt)^2} = \sqrt{v_0 ^ 2 + 2gh}\\\tan \alpha_к = \frac{v_y}{v_x} = -\frac{gt}{v_0} = - \frac{\sqrt{2gh}}{v_0}\end{aligned}\end{align} \]
Движение тела, брошенного под углом к горизонту.¶
\[ \begin{align}\begin{aligned}\begin{split}\begin{cases} v_x = v_0 * cos \alpha \\ v_y = v_0 * sin \alpha - gt \end{cases}\end{split}\\\begin{split}\begin{cases} x = v_0 \cos \alpha * t \\ y = v_0 \sin \alpha * t - \frac{gt^2}{2} \end{cases}\end{split}\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned}y = max \rightarrow v_y = 0 \rightarrow v_0 \sin \alpha -gt_{под} = 0\\V_0 = \frac{t_{под}g}{\sin \alpha}\\t_{под} = \frac{v_0 \sin \alpha}{g}\\H_{max} = y(при t =t_{под}) = v \sin \alpha t_{под} - \frac{g * t_{под} ^ 2}{2} = v_o \sin \alpha * \frac{v_0 \sin \alpha}{g} - \frac{g}{2} * \frac{v_0 ^ 2 \sin ^ 2 \alpha}{g^2} = \frac{v_0 ^ 2 \sin ^ 2 \alpha}{2g}\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned}t{пол} = 2 t{под}\\y = 0 \rightarrow v_0 \sin \alpha - \frac{gt^2}{2} \rightarrow t{пол} = \frac{2v_o \sin \alpha }{g}\\2 \sin \alpha * \cos \alpha = \sin 2 \alpha \rightarrow L = v_x * t_{пол} = v_0 \cos \alpha * \frac{2v_o \sin \alpha }{g} = \frac{v_o ^ 2 \sin 2 \alpha}{g}\\L_{max} \rightarrow 2 \alpha = \rightarrow 1 \alpha = 45\\\tan \alpha = \frac{v_y}{v_x}\end{aligned}\end{align} \]
Движение тела, брошенного горизонтально.¶
\[ \begin{align}\begin{aligned}x = x_0 + V_{0x} * t + \frac{a_x t ^ 2}{2}\\x_0=h_0, v_{0x}=0, a_x=g \rightarrow x = h - \frac{gt^2}{2}\\v_x = v_{0x}+a_xt=-gt\\x=0, h - \frac{gt_п^2}{2}=0 \rightarrow t_п = \sqrt{\frac{2a}{g}}\\v_к=-gt=-\sqrt{2gh}\end{aligned}\end{align} \]