A particle is projected with speed \(u ~\text{ms}^{–1}\) at an angle of \(\theta\) above the horizontal from a point \(O\). At time \(t ~\text{s}\) after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x ~\text{m}\) and \(y ~\text{m}\) respectively.

\((\text{i})\) Express \(x\) and \(y\) in terms of \(t\) and \(\theta\) and hence obtain the equation of trajectory \[\large y= x\tan \theta -\dfrac{gx^2 \text{sec}^2 \theta}{2u^2}\]

In a shot put competition, a shot is thrown from a height of \(2.1 \text{m}\) above horizontal ground. It has initial velocity of \(14 \text{ms}^{–1}\) at an angle of \(\theta\) above the horizontal. The shot travels a horizontal distance of \(22 \text{m}\) before hitting the ground.

\((\text{ii})\) Show that \(12.1 \tan^2 \theta - 22 \tan \theta +10=0\) , hence find \(\theta\).

\((\text{ii})\) Find the time of flight of the shot.

**Input the time of flight to three significant figures.**

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