| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2004 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Relative velocity: find resultant velocity (magnitude and/or direction) |
| Difficulty | Standard +0.3 This is a standard M4 relative velocity problem requiring vector addition and component resolution. Part (a) involves equating j-components (routine), and part (b) requires finding the magnitude of the resultant velocity vector. While it needs careful notation and understanding of relative velocity, it follows a well-established method with no novel insight required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10e Position vectors: and displacement1.10h Vectors in kinematics: uniform acceleration in vector form |
\begin{enumerate}
\item \hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively.]
\end{enumerate}
An aeroplane makes a journey from a point $P$ to a point $Q$ which is due east of $P$. The wind velocity is $w ( \cos \theta \mathbf { i } + \sin \theta \mathbf { j } )$, where $w$ is a positive constant. The velocity of the aeroplane relative to the wind is $v ( \cos \phi \mathbf { i } - \sin \phi \mathbf { j } )$, where $v$ is a constant and $v > w$. Given that $\theta$ and $\phi$ are both acute angles,\\
(a) show that $v \sin \phi = w \sin \theta$,\\
(b) find, in terms of $v , w$ and $\theta$, the speed of the aeroplane relative to the ground.\\
\hfill \mbox{\textit{Edexcel M4 2004 Q1 [6]}}