A specialized rescue apparatus at sea fires a tethered buoy from a deck towards a distress signal. The signal is located on a floating platform at a horizontal distance of LLL metres from the launch site. The buoy is projected with an initial velocity of u m s−1u \text{ m s}^{-1}u m s−1 at an elevation angle of β\betaβ to the horizontal. By modelling the buoy as a particle and the platform as being at the same horizontal level as the launch point, and neglecting air resistance, prove that for the buoy to reach at least the distance of the platform, the angle β\betaβ must satisfy the condition:
sin2β≥Lgu2 \sin 2\beta \ge \frac{Lg}{u^2} sin2β≥u2Lg