1 Two sledges P and Q, with their loads, have masses of 200 kg and 250 kg respectively and are sliding on a horizontal surface against negligible resistance. There is an inextensible light rope connecting the sledges that is horizontal and parallel to the direction of motion.

Fig. 1 shows the initial situation with both sledges travelling with a velocity of $5 \mathbf { i m ~ } \mathbf { m } ^ { - 1 }$.

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\caption{Fig. 1}
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A mechanism on Q jerks the rope so that there is an impulse of $250 \mathbf { i N s }$ on P .\\
(i) Show that the new velocity of $P$ is $6.25 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }$ and find the new velocity of $Q$.

There is now a direct collision between the sledges and after the impact P has velocity $4.5 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }$.\\
(ii) Show that the velocity of Q becomes $5.4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Calculate the coefficient of restitution in the collision.

Before the rope becomes taut again, the velocity of P is increased so that it catches up with Q . This is done by throwing part of the load from sledge P in the $- \mathbf { i }$ direction so that P 's velocity increases to $5.5 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }$. The part of the load thrown out is an object of mass 20 kg .\\
(iii) Calculate the speed of separation of the object from P .

When the sledges directly collide again they are held together and move as a single object.\\
(iv) Calculate the common velocity of the pair of sledges, giving your answer correct to 3 significant figures.

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  \includegraphics[alt={},max width=\textwidth]{1a605f0b-f595-4bb9-9624-f816c789ad86-3_987_524_258_769}
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\caption{not to scale he lengths are}
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Fig. 2

Fig. 2 shows a stand on a horizontal floor and horizontal and vertical coordinate axes $\mathrm { O } x$ and $\mathrm { O } y$. The stand is modelled as

\begin{itemize}
  \item a thin uniform rectangular base PQRS, 30 cm by 40 cm with mass 15 kg ; the sides QR and PS are parallel to $\mathrm { O } x$,
  \item a thin uniform vertical rod of length 200 cm and mass 3 kg that is fixed to the base at O , the mid-point of PQ and the origin of coordinates,
  \item a thin uniform top rod AB of length 50 cm and mass $2 \mathrm {~kg} ; \mathrm { AB }$ is parallel to $\mathrm { O } x$.
\end{itemize}

Coordinates are referred to the axes shown in the figure.\\

\hfill \mbox{\textit{OCR MEI M2 2010 Q1 [17]}}