| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2003 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Variable force (position x) - find velocity |
| Difficulty | Standard +0.3 This is a standard M3 variable force problem requiring F=ma with v dv/dx, integration of a polynomial, and applying boundary conditions. The setup is straightforward with clear given information, and the mathematical steps (integrating x(4-3x) and substituting x=6) are routine for M3 students, making it slightly easier than average. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration3.03d Newton's second law: 2D vectors6.06a Variable force: dv/dt or v*dv/dx methods |
A toy car of mass $0.2$ kg is travelling in a straight line on a horizontal floor. The car is modelled as a particle. At time $t = 0$ the car passes through a fixed point $O$. After $t$ seconds the speed of the car is $v \text{ m s}^{-1}$ and the car is at a point $P$ with $OP = x$ metres. The resultant force on the car is modelled as $\frac{1}{5}x(4 - 3x)$ N in the direction $OP$. The car comes to instantaneous rest when $x = 6$.
Find
\begin{enumerate}[label=(\alph*)]
\item an expression for $v^2$ in terms of $x$, [7]
\item the initial speed of the car. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2003 Q3 [9]}}