A particle is projected from a point with speed $u$ at an angle of elevation $α$ above the horizontal and moves freely under gravity. When it has moved a horizontal distance $x$, its height above the point of projection is $y$.

\begin{enumerate}[label=(\alph*)]
\item Show that
$$y = x \tan α - \frac{gx^2}{2u^2}(1 + \tan^2 α).$$
[5]
\end{enumerate}

A shot-putter puts a shot from a point $A$ at a height of 2 m above horizontal ground. The shot is projected at an angle of elevation of $45°$ with a speed of 14 m s$^{-1}$. By modelling the shot as a particle moving freely under gravity,

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find, to 3 significant figures, the horizontal distance of the shot from $A$ when the shot hits the ground,
[5]
\end{enumerate}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find, to 2 significant figures, the time taken by the shot in moving from $A$ to reach the ground.
[2]
\end{enumerate}

A small smooth ball $A$ of mass $m$ is moving on a horizontal table with speed $v$ when it collides directly with another small smooth ball $B$ of mass $3m$ which is at rest on the table. The balls have the same radius and the coefficient of restitution between the balls is $e$. The direction of motion of $A$ is reversed as a result of the collision.

\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $e$ and $u$, the speeds of $A$ and $B$ immediately after the collision.
[7]
\end{enumerate}

In the subsequent motion $B$ strikes a vertical wall, which is perpendicular to the direction of motion of $B$, and rebounds. The coefficient of restitution between $B$ and the wall is $\frac{1}{2}$.

Given that there is a second collision between $A$ and $B$,

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the range of values of $e$ for which the motion described is possible.
[6]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M2  Q5 [25]}}