Standard +0.8 This is a multi-part Further Maths question combining first-order linear differential equations with integrating factors, piecewise modelling, and SHM. While each technique is standard for FM students, the question requires sustained problem-solving across five parts with changing conditions, placing it moderately above average difficulty for A-level but routine for Further Maths Pure.
Show that the motion of the particle can be modelled by the differential equation
$$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$
The particle is at rest when \(t = 0\).
Find \(v\) in terms of \(t\).
Find the velocity of the particle when \(t = 2\).
When \(t = 2\) the force acting in the positive \(x\)-direction is replaced by a constant force of magnitude \(\frac { 1 } { 2 } \mathrm {~N}\) in the same direction.
Refine the differential equation given in part (a) to model the motion for \(t \geqslant 2\).
Use the refined model from part (d) to find an exact expression for \(v\) in terms of \(t\) for \(t \geqslant 2\).
\(6 \quad A\) is a fixed point on a smooth horizontal surface. A particle \(P\) is initially held at \(A\) and released from rest.
It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after 0.2 seconds at point \(B\) whose displacement is 0.2 m from \(A\). The point \(M\) is halfway between \(A\) and \(B\).
The displacement of \(P\) from \(M\) at time \(t\) seconds after release is denoted by \(x \mathrm {~m}\).
On the axes provided in the Printed Answer Booklet, sketch a graph of \(x\) against \(t\) for \(0 \leqslant t \leqslant 0.4\).
Find the displacement of \(P\) from \(M\) at 0.75 seconds after release.
\begin{enumerate}[label=(\alph*)]
\item Show that the motion of the particle can be modelled by the differential equation
$$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$
The particle is at rest when $t = 0$.
\item Find $v$ in terms of $t$.
\item Find the velocity of the particle when $t = 2$.
When $t = 2$ the force acting in the positive $x$-direction is replaced by a constant force of magnitude $\frac { 1 } { 2 } \mathrm {~N}$ in the same direction.
\item Refine the differential equation given in part (a) to model the motion for $t \geqslant 2$.
\item Use the refined model from part (d) to find an exact expression for $v$ in terms of $t$ for $t \geqslant 2$.\\
$6 \quad A$ is a fixed point on a smooth horizontal surface. A particle $P$ is initially held at $A$ and released from rest.
It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after 0.2 seconds at point $B$ whose displacement is 0.2 m from $A$. The point $M$ is halfway between $A$ and $B$.
The displacement of $P$ from $M$ at time $t$ seconds after release is denoted by $x \mathrm {~m}$.\\
(a) On the axes provided in the Printed Answer Booklet, sketch a graph of $x$ against $t$ for $0 \leqslant t \leqslant 0.4$.\\
(b) Find the displacement of $P$ from $M$ at 0.75 seconds after release.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2019 Q6 [6]}}