2 A student is studying the speed of sound, $u$, in a gas under different conditions.\\
He assumes that $u$ depends on the pressure, $p$, of the gas, the density, $\rho$, of the gas and the wavelength, $\lambda$, of the sound in the relationship $u = k p ^ { \alpha } \rho ^ { \beta } \lambda ^ { \gamma }$, where $k$ is a dimensionless constant. (The wavelength of a sound is the distance between successive peaks in the sound wave.)
\begin{enumerate}[label=(\alph*)]
\item Use the fact that density is mass per unit volume to find $[ \rho ]$.
\item Given that the units of $p$ are $\mathrm { Nm } ^ { - 2 }$, determine the values of $\alpha , \beta$ and $\gamma$.
\item Comment on what the value of $\gamma$ means about how fast sounds of different wavelengths travel through the gas.

The student carries out two experiments, $A$ and $B$, to measure $u$. Only the density of the gas varies between the experiments, all other conditions being unchanged. He finds that the value of $u$ in experiment $B$ is double the value in experiment $A$.
\item By what factor has the density of the gas in experiment $A$ been multiplied to give the density of the gas in experiment $B$ ?

Particles $A$ of mass $2 m$ and $B$ of mass $m$ are on a smooth horizontal floor. $A$ is moving with speed $u$ directly towards a vertical wall, and $B$ is at rest between $A$ and the wall (see diagram).\\
\includegraphics[max width=\textwidth, alt={}, center]{d6a0d7a6-4166-4c26-a461-39b2414c0412-03_211_795_285_244}\\
$A$ collides directly with $B$. The coefficient of restitution in this collision is $\frac { 1 } { 2 }$.\\
$B$ then collides with the wall, rebounds, and collides with $A$ for a second time.\\
(a) Show that the speed of $B$ after its second collision with $A$ is $\frac { 1 } { 2 } u$.

The first collision between $A$ and $B$ occurs at a distance $d$ from the wall. The second collision between $A$ and $B$ occurs at a distance $\frac { 1 } { 5 } d$ from the wall.\\
(b) Find the coefficient of restitution for the collision between $B$ and the wall.
\end{enumerate}

\hfill \mbox{\textit{OCR FM1 AS 2021 Q2 [11]}}