Question - Calculating Launch Angle for Projectile Motion
Solution:
The image contains a text in Italian that talks about a projectile being fired with an initial velocity, and it asks to calculate the launch angle such that the projectile can hit a target placed a certain distance away. Unfortunately, the full details of the problem are not completely visible, but I can provide a generic approach for such a problem.To solve for the launch angle ($$\theta$$) of a projectile given the initial velocity ($$v_0$$) and the distance to the target ($$R$$), we can use the following formula, assuming no air resistance and that the launch and target points are at the same vertical level:\[ R = \frac{v_0^2}{g} \sin(2 \theta) \]Where $$ R $$ is the range or distance to the target, $$ v_0 $$ is the initial velocity, $$ g $$ is the acceleration due to gravity (which is approximately $$ 9.81 m/s^2 $$ on Earth), and $$ \theta $$ is the launch angle.Rearrange this equation to solve for $$ \theta $$:\[ \sin(2 \theta) = \frac{R g}{v_0^2} \]Take the inverse sine to find $$ 2\theta $$:\[ 2\theta = \sin^{-1}\left(\frac{R g}{v_0^2}\right) \]Then, divide by 2 to find the angle $$ \theta $$:\[ \theta = \frac{1}{2} \sin^{-1}\left(\frac{R g}{v_0^2}\right) \]Now, input the given values ($$ v_0 = 130 m/s $$ and $$ R = 1100 m $$), and solve for $$ \theta $$:\[ \theta = \frac{1}{2} \sin^{-1}\left(\frac{1100 \times 9.81}{(130)^2}\right) \]\[ \theta = \frac{1}{2} \sin^{-1}\left(\frac{10791}{16900}\right) \]\[ \theta = \frac{1}{2} \sin^{-1}(0.6383) \]\[ \theta \approx \frac{1}{2} \times 39.86^\circ \]\[ \theta \approx 19.93^\circ \]The launch angle needed to hit the target 1100 meters away with an initial velocity of 130 m/s is approximately $$ 19.93^\circ $$. Keep in mind that due to the nature of the sin function there are two possible solutions for $$ \theta $$ in the range [0°, 90°], as sin(θ) = sin(180° - θ). Therefore, the other possible angle would be $$ 90^\circ - 19.93^\circ = 70.07^\circ $$. These two angles represent the low and high trajectories that both reach the target.
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