this is a problem I have encountered in my Grade 12 Maths book that I am unable to complete.
A garden sprinkler sprays water symmetrically about its vertical axis at a constant speed of V metres per second. The inital direction of the spray varies continuously between angles of $15$ and $60$ degrees to the horizontal.
Prove that, from a fixed position O on level ground, the sprinkler will wet the surface of an annular region with centre O and with internal and external radii $\frac{V²}{2g}$ and $\frac{V²}{g}$ metres respectively
I first created my 3 time-vectors (I'm not sure how to put the annotations on the "i" and "j")
$a(t)=(-g)j$
$v(t)=(V\cos{\theta})i+(-gt+V\sin{\theta})j$
$r(t)=(Vt\cos{\theta})i+(-\frac{1}{2}gt^2+Vt\sin{\theta})j$
So from these I just solved for when the j component of $r(t)=0$
$0=-\frac{1}{2}gt^2+Vt\sin{\theta}$
$t=0,\frac{2V\sin{\theta}}{g}$
I already knew about $0$ so I just substituted $60$ and $15$ into $\frac{2V\sin{\theta}}{g}$ and then into the i component of $r(t)$
$V\cos{\theta}\times\frac{2V\sin{\theta}}{g}$
$\frac{2V^2}{g}\cos{\theta}\sin{\theta}$
$\frac{2V^2}{g}\cos{15}\sin{15}=\frac{v^2}{2g}$
$\frac{2V^2}{g}\cos{60}\sin{60}=\frac{\sqrt{3}v^2}{2g}$
I am wondering why I got a correct value for 15 degrees, but not 60 degrees. Thanks in advance
Hint: 60 degrees is not the angle that gives the maximal external radius. What angle is that?