Q5 (21-SPO1U59-MECA-TS1-Q5)
Calculer la distance AS. L’objet atteint-il le point B ? Justifier en comparant AS à AB.
$$
\begin{aligned}
& x_A=0 \\
& z_A=100 \\
& A=\left(\begin{array}{c}
0 \\
100
\end{array}\right) \\
& S=\left(\begin{array}{l}
x\left(t_S\right) \\
z\left(t_S\right)
\end{array}\right)
\end{aligned}
$$
$$
\begin{aligned}
x\left(t_s\right) & =v_0 \cos (\alpha)\, t_S \\
& =80 \frac{\sqrt{3}}{2} \cdot 2 \\
& =138.5
\end{aligned}
$$
$$
\begin{aligned}
z\left(t_s\right) & =\frac{1}{2} g t_S^2+v \cdot \sin (\alpha) t_s \\
& =\frac{1}{2} 10 \cdot 2^2+\frac{80}{2} \cdot 2 \\
& =20+80 \\
& =100
\end{aligned}
$$
$$
S=\left(\begin{array}{l}
138.5 \\
100
\end{array}\right)
$$
$$
\begin{aligned}
A S & =\left(\begin{array}{l}
x_S-x_A \\
z_S-z_A
\end{array}\right) \\
& =\left(\begin{array}{c}
x_S-0 \\
100-100
\end{array}\right) \\
& =\left(\begin{array}{l}
x_S \\
0
\end{array}\right)
\end{aligned}
$$
$$
\begin{aligned}
\|\overrightarrow{A S}\| & =\sqrt{x_S^2} \\
& =x_S \\ &=138,5
\end{aligned}
$$
$$
\beta=\frac{\pi}{2}-\alpha
$$
$$
\tan \left(\frac{\pi}{2}-\alpha\right)=\frac{\|\overrightarrow{A B}\|}{\|\overrightarrow{O A}\|}
$$
$$\|\overrightarrow{A B}\|=\tan \left(\frac{\pi}{2}-\alpha\right)\|\overrightarrow{O A}\|$$
$$
\|\overrightarrow{O A}\|=100
$$
$$
\|\overrightarrow{A B}\|=100 \sqrt{3} \approx 173.2
$$
$$
\|\overrightarrow{A S}\|<\|\overrightarrow{A B}\|
$$