Moderate -0.3 This is a straightforward M2 mechanics question testing standard collision theory. Part (a) uses basic constant acceleration (F=ma, v=u+at), while parts (b) and (c) apply conservation of momentum and coefficient of restitution formulas with routine calculations. The vector collision in part (b) requires resolving components but follows standard procedures. All techniques are textbook applications with no novel problem-solving required, making it slightly easier than average for A-level.
A battering-ram consists of a wooden beam fixed to a trolley. The battering-ram runs along horizontal ground and collides directly with a vertical wall, as shown in Fig. 1.1. The battering-ram has a mass of 4000 kg.
\includegraphics{figure_1}
Initially the battering-ram is at rest. Some men push it for 8 seconds and let go just as it is about to hit the wall. While the battering-ram is being pushed, the constant overall force on it in the direction of its motion is 1500 N.
At what speed does the battering-ram hit the wall? [3]
The battering-ram hits a loose stone block of mass 500 kg in the wall. Linear momentum is conserved and the coefficient of restitution in the impact is 0.2.
Calculate the speeds of the stone block and of the battering-ram immediately after the impact. [6]
Calculate the energy lost in the impact. [3]
Small objects A and B are sliding on smooth, horizontal ice. Object A has mass 4 kg and speed 18 m s\(^{-1}\) in the \(\mathbf{i}\) direction. B has mass 8 kg and speed 9 m s\(^{-1}\) in the direction shown in Fig. 1.2, where \(\mathbf{i}\) and \(\mathbf{j}\) are the standard unit vectors.
\includegraphics{figure_2}
Write down the linear momentum of A and show that the linear momentum of B is \((36\mathbf{i} + 36\sqrt{3}\mathbf{j})\) N s. [2]
After the objects meet they stick together (coalesce) and move with a common velocity of \((u\mathbf{i} + v\mathbf{j})\) m s\(^{-1}\).
Calculate \(u\) and \(v\). [3]
Find the angle between the direction of motion of the combined object and the \(\mathbf{i}\) direction. Make your method clear. [2]
\begin{enumerate}[label=(\alph*)]
\item A battering-ram consists of a wooden beam fixed to a trolley. The battering-ram runs along horizontal ground and collides directly with a vertical wall, as shown in Fig. 1.1. The battering-ram has a mass of 4000 kg.
\includegraphics{figure_1}
Initially the battering-ram is at rest. Some men push it for 8 seconds and let go just as it is about to hit the wall. While the battering-ram is being pushed, the constant overall force on it in the direction of its motion is 1500 N.
\begin{enumerate}[label=(\roman*)]
\item At what speed does the battering-ram hit the wall? [3]
\end{enumerate}
The battering-ram hits a loose stone block of mass 500 kg in the wall. Linear momentum is conserved and the coefficient of restitution in the impact is 0.2.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Calculate the speeds of the stone block and of the battering-ram immediately after the impact. [6]
\item Calculate the energy lost in the impact. [3]
\end{enumerate}
\item Small objects A and B are sliding on smooth, horizontal ice. Object A has mass 4 kg and speed 18 m s$^{-1}$ in the $\mathbf{i}$ direction. B has mass 8 kg and speed 9 m s$^{-1}$ in the direction shown in Fig. 1.2, where $\mathbf{i}$ and $\mathbf{j}$ are the standard unit vectors.
\includegraphics{figure_2}
\begin{enumerate}[label=(\roman*)]
\item Write down the linear momentum of A and show that the linear momentum of B is $(36\mathbf{i} + 36\sqrt{3}\mathbf{j})$ N s. [2]
\end{enumerate}
After the objects meet they stick together (coalesce) and move with a common velocity of $(u\mathbf{i} + v\mathbf{j})$ m s$^{-1}$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Calculate $u$ and $v$. [3]
\item Find the angle between the direction of motion of the combined object and the $\mathbf{i}$ direction. Make your method clear. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI M2 2008 Q1 [19]}}