6

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-4_664_969_264_589}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}

A cyclist $P$ travels along a straight road starting from rest at $A$ and accelerating at $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ up to a speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. He continues at a constant speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, passing through the point $B 20 \mathrm {~s}$ after leaving $A$. Fig. 1 shows the ( $t , v$ ) graph of $P$ 's journey for $0 \leqslant t \leqslant 20$. Find\\
(i) the time for which $P$ is accelerating,\\
(ii) the distance $A B$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-4_607_937_1420_605}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}

Another cyclist $Q$ travels along the same straight road in the opposite direction. She starts at rest from $B$ at the same instant that $P$ leaves $A$. Cyclist $Q$ accelerates at $2 \mathrm {~ms} ^ { - 2 }$ up to a speed of $8 \mathrm {~ms} ^ { - 1 }$ and continues at a constant speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, passing through the point $A 20 \mathrm {~s}$ after leaving $B$. Fig. 2 shows the $( t , x )$ graph of $Q$ 's journey for $0 \leqslant t \leqslant 20$, where $x$ is the displacement of $Q$ from $A$ towards $B$.\\
(iii) Sketch a copy of Fig. 1 and add to your copy a sketch of the ( $t , v$ ) graph of $Q$ 's journey for $0 \leqslant t \leqslant 20$.\\
(iv) Sketch a copy of Fig. 2 and add to your copy a sketch of the $( t , x )$ graph of $P$ 's journey for $0 \leqslant t \leqslant 20$.\\
(v) Find the value $t$ at the instant that $P$ and $Q$ pass each other.\\
\includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-5_447_739_269_703}

The upper edge of a smooth plane inclined at $70 ^ { \circ }$ to the horizontal is joined to an edge of a rough horizontal table. Particles $A$ and $B$, of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth pulley which is fixed at the top of the smooth inclined plane. Particle $A$ is held in contact with the rough horizontal table and particle $B$ is in contact with the smooth inclined plane with the string taut (see diagram). The coefficient of friction between $A$ and the horizontal table is 0.4 . Particle $A$ is released from rest and the system starts to move.\\
(i) Find the acceleration of $A$ and the tension in the string.

The string breaks when the speed of the particles is $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(ii) Assuming $A$ does not reach the pulley, find the distance travelled by $A$ after the string breaks.\\
(iii) Assuming $B$ does not reach the ground before $A$ stops, find the distance travelled by $B$ from the time the string breaks to the time that $A$ stops.

\hfill \mbox{\textit{OCR M1 2005 Q6 [13]}}