5 A particle, \(A\), is projected vertically upwards from a point \(O\) with a speed of \(80 \mathrm {~ms} ^ { - 1 }\). One second later a second particle, \(B\), with the same mass as \(A\), is projected vertically upwards from \(O\) with a speed of \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(T\) s after the first particle is projected, the two particles collide and coalesce to form a particle \(C\).

1. Show that \(T = 3.5\).
2. Find the height above \(O\) at which the particles collide.
   \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-07_2723_33_99_22}
3. Find the time from \(A\) being projected until \(C\) returns to \(O\).
   \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-08_415_912_246_580} A particle of mass 1.2 kg is placed on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 7 } { 25 }\). The particle is kept in equilibrium by a horizontal force of magnitude \(P \mathrm {~N}\) acting in a vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the particle and the plane is 0.15 . Find the least possible value of \(P\).
   \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-08_2714_38_109_2010}
   \includegraphics[max width=\textwidth, alt={}, center]{404b5565-d76f-430e-a956-e8ce569aae6a-09_2726_35_97_20}