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\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_227_901_1352_623} Two particles \(P\) and \(Q\) have masses 0.7 kg and 0.3 kg respectively. \(P\) and \(Q\) are simultaneously projected towards each other in the same straight line on a horizontal surface with initial speeds of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). Before \(P\) and \(Q\) collide the only horizontal force acting on each particle is friction and each particle decelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The particles coalesce when they collide.

1. Given that \(P\) and \(Q\) collide 2 s after projection, calculate the speed of each particle immediately before the collision, and the speed of the combined particle immediately after the collision.
2. Given instead that \(P\) and \(Q\) collide 3 s after projection,
   (a) sketch on a single diagram the \(( t , v )\) graphs for the two particles in the interval \(0 \leqslant t < 3\),
   (b) calculate the distance between the two particles at the instant when they are projected.