The diagram shows the speed-time graph for a particle during a period of \(9 T\) seconds.
\includegraphics[max width=\textwidth, alt={}, center]{3e495748-ccb7-4c99-8387-160c4f0f9d4f-2_406_1162_319_351}
If \(T = 5\), find
the acceleration for each section of the motion,
the total distance travelled by the particle.
Sketch, for this motion,
an acceleration-time graph,
a displacement-time graph.
Calculate the value of \(T\) for which the distance travelled over the \(9 T\) seconds is 3.708 km .
Two smooth spheres \(A\) and \(B\), of masses 60 grams and 90 grams respectively, are at rest on a smooth horizontal table. \(A\) is projected towards \(B\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and the particles collide. After the collision, \(A\) and \(B\) move in the same direction as each other, with speeds \(u \mathrm {~ms} ^ { - 1 }\) and \(6 u \mathrm {~ms} ^ { - 1 }\) respectively. Calculate
the value of \(u\),
the magnitude of the impulse exerted by \(A\) on \(B\), stating the units of your answer.
(3 marks)
\(A\) and \(B\) are now replaced in their original positions and projected towards each other with speeds \(2 \mathrm {~ms} ^ { - 1 }\) and \(8 \mathrm {~ms} ^ { - 1 }\) respectively. They collide again, after which \(A\) moves with speed \(7 \mathrm {~ms} ^ { - 1 }\), its direction of motion being reversed.
Find the speed of \(B\) after this collision and state whether its direction of motion has been reversed.
In a theatre, three lights \(A , B\) and \(C\) are suspended from a horizontal beam \(X Y\) of length \(4.5 \mathrm {~m} . A\) and \(C\) are each of mass 8 kg and \(B\) is of mass 6 kg . The beam \(X Y\) is held in place by vertical ropes \(P X\) and \(Q Y\), as shown.
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In a simple mathematical model of this situation, \(X Y\) is modelled as a light rod.
Calculate the tension in each of \(P X\) and \(Q Y\).
In a refined model, \(X Y\) is modelled as a uniform rod of mass \(m \mathrm {~kg}\).
If the tension in \(P X\) is 1.5 times that in \(Q Y\), calculate the value of \(m\).