We're asked to find the time t when the projectile hits the ground with a given initial velocity and launch angle. In terms of physics, we need to find the time t when the height y is zero (there will technically be two times, one time was when the motion first started, and the second time is what we're trying to calculate--when it lands).
To find the time when it lands, we can use the equation
Deltay = v_(0y)t - 1/2g t^2
Deltay will be 0 (the change in its height), and we need to find the initial y-velocity, v_(0y). We can find this by
v_(0y) = v_0sinalpha = 18"m"/"s"sin((3pi)/4) = color(blue)(12.7"m"/"s"
Plugging in our known values (g = 9.80"m"/("s"^2)), we have
0 = (12.7"m"/"s")t - 1/2(9.80"m"/("s"^2))t^2
(4.90"m"/("s"^2))t^2 = (12.7"m"/"s")t
(4.90"m"/("s"^2))t = 12.7"m"/"s"
color(red)(t = 2.6"s"
Thus, the projectile will land after 2.6"s".
