1 When a stationary firework P of mass 0.4 kg is set off, the explosion gives it an instantaneous impulse of 16 N s vertically upwards.\\
(i) Calculate the speed of projection of P .

While travelling vertically upwards at $32 \mathrm {~ms} ^ { - 1 } , P$ collides directly with another firework $Q$, of mass 0.6 kg , that is moving directly downwards with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$, as shown in Fig. 1. The coefficient of restitution in the collision is 0.1 and Q has a speed of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ vertically upwards immediately after the collision.

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\caption{Fig. 1}
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(ii) Show that $u = 18$ and calculate the speed and direction of motion of P immediately after the collision.

Another firework of mass 0.5 kg has a velocity of $( - 3.6 \mathbf { i } + 5.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $\mathbf { i }$ and $\mathbf { j }$ are horizontal and vertical unit vectors, respectively. This firework explodes into two parts, C and D . Part C has mass 0.2 kg and velocity ( $3 \mathbf { i } + 4 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$ immediately after the explosion.\\
(iii) Calculate the velocity of D immediately after the explosion in the form $a \mathbf { i } + b \mathbf { j }$. Show that C and D move off at $90 ^ { \circ }$ to one another.\\[0pt]
[8]

\hfill \mbox{\textit{OCR MEI M2 2006 Q1 [17]}}