Recall the simplified equation for projectile motion:[br][br][center][math]y=-\frac{g}{200\cos^2\left(\theta\right)}x^2+\left(\tan\theta\right)x+y_0,[/math][br][/center]where [math]\theta[/math] denotes the angle of elevation in [b]degrees[/b], [math]y_0\ \text{m}[/math] denotes the initial height, and [math]g \approx 9.81\ \text{m}/\text{s}^2[/math] denotes the [b]gravitational constant[/b] on Earth.[br][br]We are going to fire a cannon, which is placed on the ground.
[b]Question 1. [/b]Write down the value of [math]y_0[/math].
Your cannon is located at the point with coordinates [math](0,0)[/math]. The angle of elevation is given by [math]\theta[/math], where [math]0^{\circ}<\theta<90^{\circ}[/math].
[b]Question 2. [/b]Use the GeoGebra applet above to find the range of values of [math]\theta[/math] so that your cannonball travels a horizontal distance greater than [math]8\ \text{m}[/math].
Approximately [math]26.0^{\circ}<\theta<64.0^{\circ}[/math].
[b]Question 3. [/b]Which value of [math]\theta[/math] will give the maximum horizontal distance?
[math]\theta=45^{\circ}[/math].[br][br][b]Extra explanation: [/b]The horizontal distance [math]x>0[/math] is given by solving[br][center][br][math]-\frac{g}{v^2\cos^2\theta}x^2+(\tan\theta)x=0\quad\Leftrightarrow\quad x=\frac{2v^2\cos^2\theta\tan\theta}{g}.[/math][br][/center][br]By trigonometric identities,[br][center][br][math]2\cos^2\theta\tan\theta=2\cos^2\theta\cdot\frac{\sin\theta}{\cos\theta}=2\sin\theta\cos\theta=\sin2\theta.[/math][br][/center][br]Therefore, the maximum distance is given by [math]x=\frac{v^2\sin2\theta}{g}[/math].[br][br]This distance is maximised when[br][br][center][math]\sin2\theta=1\quad\Leftrightarrow\quad2\theta=\frac{\pi}{2}\quad\Leftrightarrow\quad\theta=\frac{\pi}{4}\quad\Leftrightarrow\quad\alpha=45.[/math][br][/center]
Your territory and the enemy territory is separated by a [math]3\ \text{m}[/math]-tall wall. located [math]4\ \text{m}[/math] away from you. The land to the left of the wall belongs to your territory, while the land to the right of the wall belongs to the enemy territory.
[b]Question 4. [/b]Use the GeoGebra applet to find the range of values of [math]\theta[/math] so that your cannonball enters the enemy territory.
Approximately [math]54.0^{\circ}<\theta<73.0^{\circ}[/math].