| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Challenging +1.2 This is a vector differential equation requiring solving a second-order ODE with given initial conditions, then finding a Cartesian equation. While it involves multiple steps (solving the characteristic equation, applying two initial conditions separately for i and j components, then eliminating the parameter), the technique is standard for M5 and follows a well-practiced method. The 8+2 mark allocation reflects moderate length rather than exceptional difficulty. It's harder than routine C3/C4 calculus but typical for Further Maths mechanics modules. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors4.10d Second order homogeneous: auxiliary equation method |
A particle $P$ moves in the $x$-$y$ plane and has position vector $\mathbf{r}$ metres at time $t$ seconds. It is given that $\mathbf{r}$ satisfies the differential equation
$$\frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2} = 2\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}.$$
When $t = 0$, $P$ is at the point with position vector $3\mathbf{i}$ metres and is moving with velocity $\mathbf{j}$ m s$^{-1}$.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathbf{r}$ in terms of $t$. [8]
\item Describe the path of $P$, giving its cartesian equation. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 2006 Q3 [10]}}