Edexcel M2 2023 June — Question 7 15 marks

Exam BoardEdexcel
ModuleM2 (Mechanics 2)
Year2023
SessionJune
Marks15
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProjectiles
TypeVector form projectile motion
DifficultyStandard +0.3 This is a standard M2 projectile question using vector notation with straightforward parts: (a) uses energy conservation (routine), (b) finds angle from velocity components (standard), (c) uses kinematic equations (routine), and (d) requires finding when velocity is perpendicular using dot product (slightly less routine but still a standard technique). All parts follow well-established methods with no novel insight required, making it slightly easier than average.
Spec3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model6.02d Mechanical energy: KE and PE concepts6.02e Calculate KE and PE: using formulae

  1. \hspace{0pt} [In this question, the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52966963-2e62-4361-bcd5-a76322f8621e-24_679_1009_347_529} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A small ball is projected with velocity \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from the fixed point \(A\).
The point \(A\) is 20 m above horizontal ground.
The ball hits the ground at the point \(B\), as shown in Figure 4.
The ball is modelled as a particle moving freely under gravity.
  1. By considering energy, find the speed of the ball at the instant immediately before it hits the ground.
  2. Find the direction of motion of the ball at the instant immediately before it hits the ground.
  3. Find the time taken for the ball to travel from \(A\) to \(B\). At the instant when the direction of motion of the ball is perpendicular to ( \(3 \mathbf { i } + 2 \mathbf { j }\) ) the ball is \(h\) metres above the ground.
  4. Find the value of \(h\).

\begin{enumerate}
  \item \hspace{0pt} [In this question, the perpendicular unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in a vertical plane with $\mathbf { i }$ being horizontal and $\mathbf { j }$ being vertically upwards.]
\end{enumerate}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{52966963-2e62-4361-bcd5-a76322f8621e-24_679_1009_347_529}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

A small ball is projected with velocity $( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ from the fixed point $A$.\\
The point $A$ is 20 m above horizontal ground.\\
The ball hits the ground at the point $B$, as shown in Figure 4.\\
The ball is modelled as a particle moving freely under gravity.\\
(a) By considering energy, find the speed of the ball at the instant immediately before it hits the ground.\\
(b) Find the direction of motion of the ball at the instant immediately before it hits the ground.\\
(c) Find the time taken for the ball to travel from $A$ to $B$.

At the instant when the direction of motion of the ball is perpendicular to ( $3 \mathbf { i } + 2 \mathbf { j }$ ) the ball is $h$ metres above the ground.\\
(d) Find the value of $h$.

\hfill \mbox{\textit{Edexcel M2 2023 Q7 [15]}}