Leaving Cert Applied Maths 2013

A ball is thrown vertically upwards with a speed of \[44·1 m s ^{-1}\]       Calculate the time interval between the instants that the ball is 39·2 m above the  point of projection.  \[s = ut +\frac{1}{2}at^2\]\[39.2 = 44.1t + \frac{1}{2}(-9.8)t^2\]\[t^2 - 9t +8 = 0\]\[(t-1)(t-8) = 0\]...\[t=1, t=8\]\[t_1 = 8-1\]\[t_1 = 7s\]

An aircraft P, flying at \[600 kmh^{-1}\] sets out to intercept a second aircraft Q, which is a distance away in a direction west 30 degrees south, and flying due east at \[600 kmh^{-1}\] Find the direction in which P should fly in order to intercept Q. \[\vec V_P = -600\cos a \vec i - 600\sin a \vec j\]\[\vec V_Q = 600 \vec i\]\[\vec V_{PQ} = \vec V_P - \vec V_Q = {-600\cos a - 600}\vec i - 600\sin a \vec j\]\[\tan30 = \frac{600\sin a}{600\cos a+600}\]\[\sqrt{3} \sin a = \cos a +1 \]\[3\sin^2 a = \cos^2 a +2\cos a +1\]\[3(1 - \cos^2 a) = \cos^2 a + 2\cos a +1\]\[0=4\cos^2 a + 2\cos a -2\]\[\cos a = \frac{1}{2}\]... \[a = 60^{\circ}\]...\[W 60^{\circ} S\] or \[S 30^{\circ} W\]

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