Right, I have not done many of these questions before so I have every reason to be curfuddled.
It is Question 9 from STEP I 2004;
A particle is projected over level ground with speed u at an angle θ above the horizontal. Derive an expression for the greatest height of the particle in terms of u, θ and g
This part is fair enough
A particle is projected from the floor of a horizontal tunnel of height 9/10(d). Point P is ½d metres vertically and d metres horizontally along the tunnel from the point of projection. The particle passes through point P and lands inside the tunnel without hitting the roof. Show that;
arctan0.6 < θ < arctan3
With this I used the fact the angle will be at a maximum when the maximum height of the ball is basically the roof (and it worked with arctan3)
However, when I used the pretence that the maximum height at the minimum angle would be ½d, it did not work out using arctan0.6, instead working out at arctan1, which is a bit silly I presume.