3 A particle moves in a straight line starting from rest. The displacement \(s m\) of the particle from a fixed point \(O\) on the line at time \(t \mathrm {~s}\) is given by $$s = t ^ { \frac { 5 } { 2 } } - \frac { 15 } { 4 } t ^ { \frac { 3 } { 2 } } + 6$$ Find the value of \(s\) when the particle is again at rest.
\includegraphics[max width=\textwidth, alt={}, center]{f9e3d562-ae3c-49cc-bc92-56956d939252-06_730_1545_280_294} The velocity of a particle at time \(t \mathrm {~s}\) after leaving a fixed point \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The diagram shows a velocity-time graph which models the motion of the particle. The graph consists of 5 straight line segments. The particle accelerates to a speed of \(0.9 \mathrm {~ms} ^ { - 1 }\) in a period of 3 s , then travels at constant speed for 6 s , and then comes instantaneously to rest 1 s later. The particle then moves back and returns to rest at \(O\) at time \(T \mathrm {~s}\).

1. Find the distance travelled by the particle in the first 10 s of its motion.
2. Given that \(T = 12\), find the minimum velocity of the particle.
3. Given instead that the greatest speed of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(T\) and hence find the average speed of the particle for the whole of the motion.
   \includegraphics[max width=\textwidth, alt={}, center]{f9e3d562-ae3c-49cc-bc92-56956d939252-08_858_563_264_794} Four coplanar forces act at a point. The magnitudes of the forces are \(F \mathrm {~N} , 10 \mathrm {~N} , 50 \mathrm {~N}\) and 40 N . The directions of the forces are as shown in the diagram.