6 My cat Jeoffry has a mass of 4 kg and is sitting on rough ground near a sledge of mass 8 kg . The sledge is on a large area of smooth horizontal ice.

Initially, the sledge is at rest and Jeoffry jumps and lands on it with a horizontal velocity of $2.25 \mathrm {~ms} ^ { - 1 }$ parallel to the runners of the sledge. On landing, Jeoffry grips the sledge with his claws so that he does not move relative to the sledge in the subsequent motion.\\
(i) Show that the sledge with Jeoffry on it moves off with a speed of $0.75 \mathrm {~ms} ^ { - 1 }$.

With the sledge and Jeoffry moving at $0.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the sledge collides directly with a stationary stone of mass 3 kg . The stone may move freely over the ice. The coefficient of restitution in the collision is $\frac { 4 } { 15 }$.\\
(ii) Calculate the velocity of the sledge and Jeoffry immediately after the collision.

In a new situation, Jeoffry is initially sitting at rest on the sledge when it is stationary on the ice. He then walks from the back to the front of the sledge.\\
(iii) Giving a brief reason for your answer, describe what happens to the sledge during his walk.

Jeoffry is again sitting on the sledge when it is stationary on the ice. He jumps off and, after he has lost contact with the sledge, has a horizontal speed relative to the sledge of $3 \mathrm {~ms} ^ { - 1 }$.\\
(iv) Determine the speed of the sledge after Jeoffry loses contact with it.

Fig. 7 shows a container for flowers which is a vertical cylindrical shell with a closed horizontal base. Its radius and its height are both $\frac { 1 } { 2 } \mathrm {~m}$. Both the curved surface and the base are made of the same thin uniform material. The mass of the container is $M \mathrm {~kg}$.

\begin{figure}[h]
\begin{center}
  \includegraphics[width=0.8\textwidth]{54711a46-83ce-4fb9-b6d3-53b264725c74-6_323_709_447_767}
\caption{Fig. 7}
\end{center}
\end{figure}

(i) Find, as a fraction, the height above the base of the centre of mass of the container.

The container would hold $\frac { 3 } { 2 } M \mathrm {~kg}$ of soil when full to the top.

Some soil is put into the empty container and levelled with its top surface $y \mathrm {~m}$ above the base. The centre of mass of the container with this much soil is zm above the base.\\
(ii) Show that $z = \frac { 1 + 9 y ^ { 2 } } { 6 ( 1 + 3 y ) }$.\\
(iii) It is given that $\frac { \mathrm { d } z } { \mathrm {~d} y } = 0$ when $y = 0.14$ (to 2 significant figures) and that $\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} y ^ { 2 } } > 0$ at this value of $y$.

When putting in the soil, how might you use this information if the container is to be placed on slopes without it tipping over?

\section*{END OF QUESTION PAPER}

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\hfill \mbox{\textit{OCR MEI Further Mechanics Minor  Q6 [14]}}