# A projectile is shot at an angle of pi/4  and a velocity of  6 ms^-1. How far away will the projectile land?

Mar 25, 2018

$3.67 \setminus m$ (3 sf)

#### Explanation:

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

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

Horizontal Motion

The projectile will move under constant speed (NB we can still use "suvat" equation with $a = 0$).

Suppose that the projectile will travel a distance $x \setminus m$ in time $T \setminus s$, we must resolve the initial speed in the horizontal direction:

 { (s=,x,m), (u=,6 cos (pi/4)=3sqrt(2),ms^-1), (v=,"Not Required",ms^-1), (a=,0,ms^-2), (t=,T,s) :}

So applying $s = u t + \frac{1}{2} a {t}^{2}$ we get

$x = 3 \sqrt{2} T$

Vertical Motion

The projectile travels under constant acceleration due to gravity. Its displacement will be $0$ at the point where the projectile is at ground level, leading to two solution $t = 0$ (trivial) and $t = T$. Considering upwards as positive, and again resolving the initial speed (vertically this time):

 { (s=,0,m), (u=,6 sin (pi/4)=3sqrt(2),ms^-1), (v=,0,ms^-1), (a=,-g,ms^-2), (t=,T,s) :}

Applying $s = u t + \frac{1}{2} a {t}^{2}$ we have:

$0 = 3 \sqrt{2} T + \frac{1}{2} \left(- g\right) {T}^{2}$
$\therefore 0 = 6 \sqrt{2} T - g {T}^{2}$
$\therefore T \left(6 \sqrt{2} - g T\right) = 0$

Leading to two possible solution:

$\left\{\begin{matrix}T = 0 & \null & \text{(trivial solution)} \\ 6 \sqrt{2} - g T = 0 & \null & \null\end{matrix}\right.$

Hence we have:

$6 \sqrt{2} - g T = 0 \implies T = \frac{6 \sqrt{2}}{g}$

And using the result we established from the horizontal motion, we have:

$x = 3 \sqrt{2} \cdot \frac{6 \sqrt{2}}{g}$
$\setminus \setminus = \frac{36}{g}$

If we use $g = 9.8 \setminus m {s}^{- 2}$ we get:

$x = 3.67 \setminus m$ (3 sf)