# A projectile is shot at an angle of pi/12  and a velocity of 88 m/s. How far away will the projectile land?

Dec 10, 2016

For Physics or Mechanics you should learn the "suvat" equations for motion under constant acceleration:

$\left.\begin{matrix}v = u + a t & \text{ where " & s="displacement "(m) \\ s=ut+1/2at^2 & \null & u="initial speed "(ms^-1) \\ s=1/2(u+v)t & \null & v="final speed "(ms^-1) \\ v^2=u^2+2as & \null & a="acceleration "(ms^-2) \\ s=vt-1/2at^2 & \null & t="time } \left(s\right)\end{matrix}\right.$

Vertical Motion
Motion under constant acceleration due to gravity, applied vertically upwards

Let the total time that the projectile takes to its return to the ground after launch be $T$, We would expect two solutions as obviously T=0 is one solution. The total displacement will be zero.

$\left\{\begin{matrix}s = & 0 & m \\ u = & 88 \sin \left(\frac{\pi}{12}\right) & m {s}^{-} 1 \\ v = & \text{not required} & m {s}^{-} 1 \\ a = & - g & m {s}^{-} 2 \\ t = & T & s\end{matrix}\right.$

So we can calculate $T$ using $s = u t + \frac{1}{2} a {t}^{2}$

$\therefore 0 = 88 \sin \left(\frac{\pi}{12}\right) T + \frac{1}{2} \left(- g\right) {T}^{2}$
$\therefore \frac{1}{2} g {T}^{2} - 88 \sin \left(\frac{\pi}{12}\right) T = 0$
$\therefore T \left(\frac{g T}{2} - 88 \sin \left(\frac{\pi}{12}\right)\right) = 0$
$\therefore T = 0 , T = \frac{176 \sin \left(\frac{\pi}{12}\right)}{g}$

Horizontal Motion
Under constant speed (NB we can still use "suvat" equation with a=0). Thje projectile will be in in the air for the same time, $t = T$

So we can calculate the horizontal displacement $s$ using $s = u t$

$s = 88 \cos \left(\frac{\pi}{12}\right) T$
$\therefore s = \frac{88 \cos \left(\frac{\pi}{12}\right) \left(176 \sin \left(\frac{\pi}{12}\right)\right)}{g}$
$\therefore s = \frac{176 \cdot 44 \cdot 2 \cos \left(\frac{\pi}{12}\right) \sin \left(\frac{\pi}{12}\right)}{g}$
$\therefore s = \frac{7744 \sin \left(\frac{2 \pi}{12}\right)}{g}$
$\therefore s = \frac{\left(7744\right) \left(0.5\right)}{g}$
$\therefore s = \frac{3872}{g}$

So using $g = 9.8 m {s}^{-} 2$ we have.

$s = \frac{3872}{9.8} = 395.102 \ldots = 395 m$