A boy stands $5$ m from a house and throws a ball towards an open window, the sill of which is $6$ m above the point of projection. He projects the ball with speed $\backslash (13\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ at an angle of $\backslash (45^\{\backslash$circ$\}\backslash )$ to the horizontal in a plane at right angles to the window.

$($a$)$ Using the equation of the path of a projectile, or otherwise, find at what distance below the sill the ball strikes the wall.

$($b$)$ If the ball had been projected by the boy at $\backslash (14\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ from the same position, find correct to one decimal place, the two possible angles of projection for the ball to just clear the sill.

$($c$)$ State two assumptions that are made in the modelling of the particle motion of the ball