Standard +0.3 This is a standard M2 projectiles question requiring trajectory equation derivation (bookwork), then applying it to find angle constraints and calculating speed/range. Part (a) is routine derivation, (b) requires substitution and solving a quadratic inequality, (c-d) are straightforward applications of standard formulae. Slightly easier than average due to methodical structure and standard techniques throughout.
A particle \(P\) is projected from a fixed point \(O\) on horizontal ground. The particle is projected with speed \(u\) at an angle \(\alpha\) above the horizontal. At the instant when the horizontal distance of \(P\) from \(O\) is \(x\), the vertical distance of \(P\) above the ground is \(y\). The motion of \(P\) is modelled as that of a particle moving freely under gravity.
Show that \(y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)\)
(6)
A small ball is projected from the fixed point \(O\) on horizontal ground. The ball is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\theta ^ { \circ }\) above the horizontal. A vertical pole \(A B\), of height 2 m , stands on the ground with \(O A = 10 \mathrm {~m}\), as shown in Figure 3.
\begin{figure}[h]
\end{figure}
The ball is modelled as a particle moving freely under gravity and the pole is modelled as a rod.
The path of the ball lies in the vertical plane containing \(O , A\) and \(B\).
Using the model,
find the range of values of \(\theta\) for which the ball will pass over the pole.
Given that \(\theta = 40\) and that the ball first hits the ground at the point \(C\)
find the speed of the ball at the instant it passes over the pole,
find the distance \(O C\).
\includegraphics[max width=\textwidth, alt={}, center]{0762451f-b951-4d66-9e01-61ecb7b30d95-28_2649_1898_109_169}
\begin{enumerate}
\item A particle $P$ is projected from a fixed point $O$ on horizontal ground. The particle is projected with speed $u$ at an angle $\alpha$ above the horizontal. At the instant when the horizontal distance of $P$ from $O$ is $x$, the vertical distance of $P$ above the ground is $y$. The motion of $P$ is modelled as that of a particle moving freely under gravity.\\
(a) Show that $y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)$\\
(6)
\end{enumerate}
A small ball is projected from the fixed point $O$ on horizontal ground. The ball is projected with speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at angle $\theta ^ { \circ }$ above the horizontal. A vertical pole $A B$, of height 2 m , stands on the ground with $O A = 10 \mathrm {~m}$, as shown in Figure 3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-24_246_899_840_525}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
The ball is modelled as a particle moving freely under gravity and the pole is modelled as a rod.\\
The path of the ball lies in the vertical plane containing $O , A$ and $B$.\\
Using the model,\\
(b) find the range of values of $\theta$ for which the ball will pass over the pole.
Given that $\theta = 40$ and that the ball first hits the ground at the point $C$\\
(c) find the speed of the ball at the instant it passes over the pole,\\
(d) find the distance $O C$.
\includegraphics[max width=\textwidth, alt={}, center]{0762451f-b951-4d66-9e01-61ecb7b30d95-28_2649_1898_109_169}
\hfill \mbox{\textit{Edexcel M2 2022 Q7 [16]}}