Moderate -0.3 This is a standard multi-part mechanics question requiring straightforward application of conservation of momentum, coefficient of restitution formula, and projectile motion. Part (b) requires algebraic manipulation to show a given result, but follows directly from energy formulas. All techniques are routine for M2 level with no novel problem-solving required, making it slightly easier than average.
An object P , with mass 6 kg and speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is sliding on a smooth horizontal table. Object P explodes into two small parts, Q and \(\mathrm { R } . \mathrm { Q }\) has mass 4 kg and R has mass 2 kg and speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of motion of P before the explosion. This information is shown in Fig. 1.1.
\begin{figure}[h]
Calculate the velocity of Q .
Just as object R reaches the edge of the table, it collides directly with a small object S of mass 3 kg that is travelling horizontally towards R with a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). This information is shown in Fig. 1.2. The coefficient of restitution in this collision is 0.1 .
\begin{figure}[h]
Calculate the velocities of R and S immediately after the collision.
The table is 0.4 m above a horizontal floor. After the collision, R and S have no contact with the table.
Calculate the distance apart of R and S when they reach the floor.
A particle of mass \(m \mathrm {~kg}\) bounces off a smooth horizontal plane. The components of velocity of the particle just before the impact are \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the plane and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the plane. The coefficient of restitution is \(e\).
Show that the mechanical energy lost in the impact is \(\frac { 1 } { 2 } m v ^ { 2 } \left( 1 - e ^ { 2 } \right) \mathrm { J }\).
1
\begin{enumerate}[label=(\alph*)]
\item An object P , with mass 6 kg and speed $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, is sliding on a smooth horizontal table. Object P explodes into two small parts, Q and $\mathrm { R } . \mathrm { Q }$ has mass 4 kg and R has mass 2 kg and speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction of motion of P before the explosion. This information is shown in Fig. 1.1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-2_346_1267_429_479}
\captionsetup{labelformat=empty}
\caption{Fig. 1.1}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Calculate the velocity of Q .
Just as object R reaches the edge of the table, it collides directly with a small object S of mass 3 kg that is travelling horizontally towards R with a speed of $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. This information is shown in Fig. 1.2. The coefficient of restitution in this collision is 0.1 .
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-2_506_647_1215_790}
\captionsetup{labelformat=empty}
\caption{Fig. 1.2}
\end{center}
\end{figure}
\item Calculate the velocities of R and S immediately after the collision.
The table is 0.4 m above a horizontal floor. After the collision, R and S have no contact with the table.
\item Calculate the distance apart of R and S when they reach the floor.
\end{enumerate}\item A particle of mass $m \mathrm {~kg}$ bounces off a smooth horizontal plane. The components of velocity of the particle just before the impact are $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ parallel to the plane and $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ perpendicular to the plane. The coefficient of restitution is $e$.
Show that the mechanical energy lost in the impact is $\frac { 1 } { 2 } m v ^ { 2 } \left( 1 - e ^ { 2 } \right) \mathrm { J }$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI M2 2010 Q1 [17]}}