| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2004 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Variable force (position x) - find velocity |
| Difficulty | Challenging +1.2 This M3 question requires setting up F=ma with variable resistance (mx²), using v dv/dx = a, then integrating to find v². Part (a) is straightforward substitution after integration. Part (b) requires setting v=0 and solving a cubic equation. While it involves multiple steps and the v dv/dx technique specific to M3, the setup is standard for this topic and the mathematics (integration of polynomials, solving x³=15) is routine for this level. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration6.06a Variable force: dv/dt or v*dv/dx methods |
3. A particle $P$ of mass $m \mathrm {~kg}$ slides from rest down a smooth plane inclined at $30 ^ { \circ }$ to the horizontal. When $P$ has moved a distance $x$ metres down the plane, the resistance to the motion of $P$ from non-gravitational forces has magnitude $m x ^ { 2 }$ newtons.
Find
\begin{enumerate}[label=(\alph*)]
\item the speed of $P$ when $x = 2$,
\item the distance $P$ has moved when it comes to rest for the first time.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2004 Q3 [10]}}