5

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{fcd2b219-d9b4-4972-b8fe-25cf543b9054-3_245_335_580_906}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}

A force, whose direction is upwards parallel to a line of greatest slope of a plane inclined at $35 ^ { \circ }$ to the horizontal, acts on a box of mass 15 kg which is at rest on the plane. The normal component of the contact force on the box has magnitude $R$ newtons (see Fig. 1).\\
(i) Show that $R = 123$, correct to 3 significant figures.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{fcd2b219-d9b4-4972-b8fe-25cf543b9054-3_369_1045_1247_555}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}

When the force parallel to the plane acting on the box has magnitude $X$ newtons the box is about to move down the plane, and when this force has magnitude $5 X$ newtons the box is about to move up the plane (see Fig. 2).\\
(ii) Find the value of $X$ and the coefficient of friction between the box and the plane.\\
(i) A particle $P$ of mass 1.2 kg is released from rest at the top of a slope and starts to move. The slope has length 4 m and is inclined at $25 ^ { \circ }$ to the horizontal. The coefficient of friction between $P$ and the slope is $\frac { 1 } { 4 }$. Find
\begin{enumerate}[label=(\alph*)]
\item the frictional component of the contact force on $P$,
\item the acceleration of $P$,
\item the speed with which $P$ reaches the bottom of the slope.\\
(ii) After reaching the bottom of the slope, $P$ moves freely under gravity and subsequently hits a horizontal floor which is 3 m below the bottom of the slope.\\
(a) Find the loss in gravitational potential energy of $P$ during its motion from the bottom of the slope until it hits the floor.\\
(b) Find the speed with which $P$ hits the floor.\\[0pt]
[1]\\[0pt]
[3]\\
$7 \quad$ A particle $P$ starts to move from a point $O$ and travels in a straight line. At time $t$ s after $P$ starts to move its velocity is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $v = 0.12 t - 0.0006 t ^ { 2 }$.
\begin{enumerate}[label=(\roman*)]
\item Verify that $P$ comes to instantaneous rest when $t = 200$, and find the acceleration with which it starts to return towards $O$.
\item Find the maximum speed of $P$ for $0 \leqslant t \leqslant 200$.
\item Find the displacement of $P$ from $O$ when $t = 200$.
\item Find the value of $t$ when $P$ reaches $O$ again.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{CAIE M1 2002 Q5 [8]}}