Two smooth spheres \(X\) and \(Y\), of masses \(x \mathrm {~kg}\) and \(y \mathrm {~kg}\) respectively, are free to move in a smooth straight groove in a horizontal table. \(X\) is projected with speed \(6 \mathrm {~ms} ^ { - 1 }\) towards \(Y\), which is stationary. After the collision \(X\) moves with speed \(2 \mathrm {~ms} ^ { - 1 }\) and \(Y\) moves with speed \(3 \mathrm {~ms} ^ { - 1 }\).
Calculate the two possible values of the ratio \(x : y\).
State a modelling assumption that you have made concerning \(X\) and \(Y\).
\(Y\) now strikes a vertical barrier and rebounds along the groove with speed \(k \mathrm {~ms} ^ { - 1 }\), colliding again with \(X\) which is still moving at \(2 \mathrm {~ms} ^ { - 1 }\). Given that in this impact \(Y\) is brought to rest and the direction of motion of \(X\) is reversed,
show that \(k > 1 \cdot 5\).
Two particles \(P\) and \(Q\), of masses 3 kg and 2 kg respectively, rest on the smooth faces of a wedge whose cross-section is a triangle with angles \(30 ^ { \circ } , 60 ^ { \circ }\) and \(90 ^ { \circ }\), as shown. \(P\) and \(Q\) are connected by a light
\includegraphics[max width=\textwidth, alt={}, center]{f8386a80-e428-43a7-acc8-f7ab11b2a53a-2_255_607_1078_1311}
string, parallel to the lines of greatest slope of the two planes, which passes over a fixed pulley at the highest point of the wedge.
The system is released from rest with \(P 0.8 \mathrm {~m}\) from the pulley and \(Q 1 \mathrm {~m}\) from the bottom of the wedge, and \(Q\) starts to move down. Calculate
the acceleration of either particle,
the tension in the string,
the speed with which \(P\) reaches the pulley.
Two modelling assumptions have been made about the string and the pulley.
State these two assumptions and briefly describe how you have used each one in your solution.