Standard +0.8 This M5 question requires solving a second-order vector differential equation with given initial conditions. While the equation itself is straightforward (constant coefficients, homogeneous), students must recognize it as a vector ODE, find the characteristic equation (giving r=0,2), apply the general solution form, and use two initial conditions (position and velocity) to determine constants. The vector nature and the need to handle both components systematically makes this moderately challenging, though it follows standard M5 techniques without requiring novel insight.
A particle moves in a plane in such a way that its position vector \(\mathbf{r}\) metres at time \(t\) seconds satisfies the differential equation
$$\frac{d^2\mathbf{r}}{dt^2} - 2\frac{d\mathbf{r}}{dt} = 0$$
When \(t = 0\), the particle is at the origin and is moving with velocity \((4i + 2j)\) m s\(^{-1}\).
Find \(\mathbf{r}\) in terms of \(t\).
[7]
A particle moves in a plane in such a way that its position vector $\mathbf{r}$ metres at time $t$ seconds satisfies the differential equation
$$\frac{d^2\mathbf{r}}{dt^2} - 2\frac{d\mathbf{r}}{dt} = 0$$
When $t = 0$, the particle is at the origin and is moving with velocity $(4i + 2j)$ m s$^{-1}$.
Find $\mathbf{r}$ in terms of $t$.
[7]
\hfill \mbox{\textit{Edexcel M5 Q1 [7]}}